category: literaturenote citekey: almibalancedquasistaticevolutionscritical2025 title: Balanced quasistatic evolutions of critical points in metric spaces authors: "Almi, Stefano; Fornasier, Massimo; Klemenc, Jona; Scagliotti, Alessandro" year: 2025 date: 2025-00-00 2025 url: "https://arxiv.org/abs/2506.09812v1" zotero_key: QJ3PGIAZ zotero_storage: VUAEVCAX collections: transpordi_planeerimine folder: 001_artiklid firstAuthor: "Almi, Stefano"

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BALANCED QUASISTATIC EVOLUTIONS OF CRITICAL POINTS IN METRIC SPACES

STEFANO ALMI¹ , MASSIMO FORNASIER²,3,⁴ , JONA KLEMENC²,³ AND ALESSANDRO SCAGLIOTTI²,³

Abstract. Quasistatic evolutions of critical points of time-dependent energies exhibit piecewise smooth behavior, making them useful for modeling continuum mechanics phenomena like elastic-plasticity and fracture. Traditionally, such evolutions have been derived as vanishing viscosity and inertia limits, leading to balanced viscosity solutions. However, for nonconvex energies, these constructions have been realized in Euclidean spaces and assume non-degenerate critical points. In this paper, we take a different approach by decoupling the time scales of the energy evolution and of the transition to equilibria. Namely, starting from an equilibrium configuration, we let the energy evolve, while keeping frozen the system state; then, we update the state by freezing the energy, while letting the system transit via gradient flow or an approximation of it (e.g., minimizing movement or backward differentiation schemes). This approach has several advantages. It aligns with the physical principle that systems transit through energy-minimizing steady states. It is also fully constructive and computationally implementable, with physical and computational costs governed by appropriate action functionals. Additionally, our analysis is simpler and more general than previous formulations in the literature, as it does not require non-degenerate critical points. Finally, this approach extends to evolutions in locally compact metric path spaces, and our axiomatic presentation allows for various realizations.

1. Introduction

Quasistatic evolutions of critical points driven by time-dependent energies are characterized by piecewise smooth behaviors. For this reason, they have been used to model time-dependent phenomena in continuum mechanics, such as linearly elastic perfect plasticity or cohesive and brittle fracture [10, 11, 12, 13]. Traditionally, proposed constructions have been mostly relying on a vanishing viscosity and inertia limit, yielding solutions characterized by an energy balance, known as balanced viscosity solutions [28, 1, 2, 24, 25, 26] (see Section 1.1 for more details); these derivations for quite general nonconvex energies have been developed in Euclidean spaces and under the restrictive assumption of non-degeneracy of critical points of the energy. In such constructions, two

¹ Dipartimento di Matematica e Applicazioni "R. Caccioppoli", Universita di Napoli ` Federico II, via Cintia, 80126 Napoli, Italy

²CIT School, Technical University of Munich, Garching bei Munchen, Germany. ¨

³Munich Center for Machine Learning (MCML), Munich, Germany.

⁴Munich Data Science Institute (MDSI), Technical University of Munich, Garching bei Munchen, Germany ¨

E-mail addresses: stefano.almi@unina.it, massimo.fornasier@cit.tum.de, jona.klemenc@tum.de, scag@ma.tum.de.

time-scales coexist: One corresponding to the evolution of the driving energy, and the other corresponding to the aspiration of the system towards equilibria.

In this paper, we follow an alternative route, which disentangles the two time-scales. Starting from an equilibrium configuration, we define the discrete quasistatic evolution by letting the energy evolve, while keeping frozen the system state. After a small amount of time δ > 0, the state is in general out of equilibrium. We amend this situation by freezing the energy, while letting the system transit via gradient flow—or some discrete approximation of it, like minimizing movement or backward differentiation scheme. Afterwards, we iterate this procedure for the whole evolution horizon, obtaining a piecewise constant curve t 7→ η δ (t). For vanishing time discretizations δ → 0, it turns out that the limiting trajectories indeed enjoy the same energy balance properties as balanced viscosity solutions.

There are multiple advantages of our approach, which was introduced and explored first for time-evolving constraints in [8] (see [4, 5, 6, 9, 14] for some applications to phase-field fracture evolution). First of all, it follows the physical principle that a system moves through action-minimizing transitions towards a steady state. Secondly, it allows us to effortlessly combine a discrete time scale for the system transition—e.g., when considering the minimizing movement scheme—with a continuous time scale for the energy evolution. Third, the separation of the time scales leads to a disentanglement of the technical-proving challenges shared between previous approaches and ours: Taking [2] as a seminal example, the two careful arguments concerning functional compactness and trajectory surgery coexist in the proof of [2, Proposition 4.1 and Proposition 4.5]. In comparison, when deriving the limiting curve, we retrieve compactness by relying on the properties of the system transitions in a purely axiomatic way. After having applied the functional compactness arguments, we establish the validity of the axioms by trajectory surgery, in a separate step. The resulting framework leaves open the possibility of further system transitions beyond the three which we have studied here, i.e., gradient flow, minimizing movement and backward differentiation scheme (see below for more details). Lastly, our construction immediately suggests a numerical implementation approach, whereas constructions through viscosity and inertia limits face the numerical difficulties associated with exploding velocity fields.

Let us now introduce the results of the paper more formally. In the setting of this paper, we consider a locally compact metric path space (X, d), and a sufficiently wellbehaved energy ^E : [0, T]×^X ^→ ^R in charge of driving the system. We seek to construct a balanced quasistatic evolution ^ηˆ: [0, T] ^→ ^X. Being a "quasistatic evolution", we demand that

$$|\partial E_t|(\hat{\eta}(t)) = 0 \quad \text{for every } t \in [0, T] \setminus J,$$ (1.1)

where J = {t ∈ [0, T] : ˆη ⁺(t) ̸= ˆ^η ⁻(t)} is the countable jump set of ˆη. Moreover, the property of being "balanced" refers to complying with the energy balance

$$E_s(\hat{\eta}^-(s)) - E_t(\hat{\eta}^+(t)) = \int_s^t \partialt E\tau(\hat{\eta}(\tau)) d\tau - \bar{\mu}([s, t]), \quad \forall s < t \in (0, T),$$ (1.2)

for a suitable positive Radon measure ^µ ∈ M+([0, T]) supported on ^J, together with the fact that, for each t ∈ J, we have that

$$\mu({t}) = E_t(\hat{\eta}^-(t)) - E_t(\hat{\eta}^+(t)) = c_t(\hat{\eta}^-(t), \hat{\eta}^+(t)), \tag{1.3}$$

Here, c^t : ^X×^X ^→ ^R is defined by minimizing energy-dissipation integrals, and is tailored on the energy landscape at time t and on the system transition rule. Furthermore, (1.3) enforces that the balanced quasistatic evolution ˆη does not jump over energy barriers. On this point, the work [8] provided a construction of quasistatic evolutions in presence of a fixed energy and a time-varying linear constraint. Unfortunately, in that case it was not possible to achieve the energy balance. Instead, the authors established an inequality of the form

$$E_t(\hat{\eta}^+(t)) \le E_s(\hat{\eta}^-(s)) + \int_s^t \partialt E\tau(\hat{\eta}(\tau)) d\tau$$ (1.4)

for every 0 < s ≤ t < T. In contrast, we bring to completion the program outlined above, even in the presence of degenerate critical points. For such degenerate critical points not to interfere with the limiting construction, we have to set this limiting construction in a quotient space ^X of the path space [0, T] ^× ^X, where we identify points in the same connected component of the set of critical points of E^t , with the limiting curve ˆη taking values in X . In the simplified theorem, we assume Assumptions 1 to 8, which ensures that all the functions E, |∂E|, ∂tE and c all factor through X ; we denote the resulting maps as ^Eˆ, ^|∂Eˆ|, ^∂tE^ˆ and ˆ^c (Figure 1).

$$\begin{array}{cccccccccc$$

Figure 1: The spaces involved in the definition of the quasistatic evolution ˆη. If E does not have degenerate critical points, both q is the identitity and ˆη takes values in [0, T] ^× ^X.

Theorem 1 (simplified). Let us assume Assumptions 1 to 8. Furthermore, let the family of mappings ω¯^t : ^X ^→ ^X indexed by ^t ^∈ [0, T] be the transition rule as in Defini tion 1, corresponding to an action c^t and complying with Axioms 1, 2 and 3'. Then, for any positive vanishing sequence (δn)n∈^N and for the corresponding discrete quasistatic evolutions η ^δⁿ , we can—without relabeling—extract a subsequence such that:

The compositions q (id × η ^δⁿ ) converge pointwise to a piecewise continuous limiting curve ηˆ: [0, T] → X .
There exists a positive Radon measure µ¯ ∈ M([0, T]) such that µ δ ⁿ ⇀^∗ µ¯, where µ ^δⁿ is as defined in Equation (3.3). Moreover, the set J := supp ¯µ consists of countably many points.
The left and right limits ηˆ −(t) and ηˆ ⁺(t) of ^η^ˆ exist for every ^t ^∈ (0, T), and so do the limits ηˆ ⁺(0) and ηˆ −(T).
The limiting curve ηˆ fulfills, for all 0 ≤ s < t ≤ T, the energy balance identity

$$\hat{E}_s(\hat{\eta}^-(s)) - \hat{E}_t(\hat{\eta}^+(t)) = \int_s^t \partial_t \hat{E}(\hat{\eta}(\tau)) d\tau - \bar{\mu}([s,t]).$$

• ^|∂E^ˆ t | ηˆ(t) = 0 for all t ∈ [0, T]. • The limiting curve ηˆ is continuous in [0, T] \ J, and for every t ∈ J we have that Eˆ ^t(ˆη ⁻(t)) ⁻ ^E^ˆ ^t(ˆη ⁺(t)) = ¯µ({t}) = ˆc^t ηˆ −(t), ηˆ ⁺(t)

As is apparent in the theorem, we are pursuing an axiomatic construction. Comparing Axioms 1 and 2 with Axiom 3', the latter looks technically more involved—indeed, it distills the aforementioned argument of trajectory surgery—, constituting a potential obstruction in finding complying transition rules. To remedy this difficulty, we propose the notion of actions generated by curves in Definition 3 and we prove that it suffices for conforming to Axiom 3'. Moreover, we provide three examples of transition rules and their corresponding actions, namely the gradient flow (GF), the minimizing movement scheme (MMS) and the backward differentiation formula (BDF2). While the variational nature of the gradient flow is well established [7] and indeed used, e.g., in [2], the formulation of the action for MMS has been introduced in [25]. Finally, we propose here an action for the BDF2 scheme that is, to the best of our knowledge, novel. Our analysis of the BDF2 is inspired by [16], which is the first work describing the BDF2 as a numerical algorithm for the approximation of gradient flows in a metric space. By proving that those three actions are actually generated by curves, we guarantee that GF, MMS and BDF2 are suitable for constructing balanced quasistatic evolutions. However, we would like to emphasize the distinction between these actions. While the action associated with the GF (gradient flow) models a time-continuous transition, those of MMS and BDF2 are based on purely algorithmic (discrete-time) iterations, see Figure 2. Therefore, we understand the GF action as "physical" and those of MMS and BDF2 as "computational".

Finally, in Section 6, we illustrate our theoretical findings with a simple numerical experiment on a model of elastic rod fracture. Despite its simplicity, the purpose of this example is to demonstrate the full computability of our approach and its correspondence to physical principles.

Figure 2: We display here a scheme explaining how a quasistatic evolution of critical point is constructed through MMS transition rule.

1.1. Related work. Our results are inspired by the research initiated by C. Zanini in [28]. In that work, the author constructs unique evolutions of critical points of smooth, nonconvex energy functions, where the critical points satisfy certain transversality conditions. These evolutions are obtained as limits of vanishing viscosity solutions to the singularly perturbed gradient flow equation:

$$\varepsilon \dot{x}(t) = -\nabla_x E_t(x(t)).$$

The construction of these solutions involves patching together smooth branches of equilibrium solutions (slow dynamics) with heteroclinic solutions of the gradient flow (fast dynamics).

Building upon the concept of balanced viscosity solutions introduced in the context of rate-independent evolutions [17, 18, 19], V. Agostiniani and R. Rossi further proved in [2] that vanishing viscosity evolutions of isolated critical points fulfill an energy balance equations as in (1.2). We also report that in [13] quasitatic evolutions were constructed as stochastic processes obtained as limits of vanishing viscosity solutions in the framework Young measures.

In [25] G. Scilla and F. Solombrino perform a convergence analysis of a discretein-time minimization scheme approximating a finite dimensional singularly perturbed gradient flow. They allow for different scalings between the viscosity parameter ε and the time scale τ . When the ratio ε/τ diverges, the authors prove the convergence of this scheme to balanced viscosity solutions of the quasistatic evolution problem obtained as a formal limit for ε → 0 of the gradient flow. They also characterize the limit evolution corresponding to an asymptotically finite ratio between the scales, and they derived the expression of the transition action c^t corresponding to the minimizing movement scheme.

Although transversality conditions for critical points as in [28] are known to be generically fulfilled [3], they exclude interesting situations, often appearing in the application (for instance, stationary solutions to (1.1) whose stability changes depending on the time t, usually giving rise to bifurcation of other branches of critical points). In [24], G. Scilla and F. Solombrino investigate the phenomenon of delayed loss of stability in singularly perturbed gradient flows. Their study focuses on the relaxation of transversality conditions imposed on critical points, exploring the consequences of their removal.

The papers [1, 26] also study the vanishing inertia and viscosity limit of a second order system set, driven by a possibly nonconvex time-dependent energy satisfying very general assumptions. By means of a variational approach, they show that the solutions of the singularly perturbed problem converge to a curve of stationary points of the energy and characterize the behavior of the limit evolution at jump times. At those times, the left and right limits of the evolution are connected by a finite number of heteroclinic solutions to the unscaled equation.

To position our contribution within this line of research, we emphasize that the aforementioned works [28, 2, 25, 24] have exclusively studied the vanishing viscosity limit of singularly perturbed gradient flows in Euclidean spaces. In contrast, this paper extends the construction of balanced quasistatic evolutions of critical points to metric spaces, without imposing transversality conditions or assuming isolated critical points1 .

¹From private communications by G. Savar´e, we are aware that V. Agostiniani, R. Rossi, and G. Savar´e have been working for a while on generalizing the vanishing viscosity limit of solutions to singularly perturbed gradient flows in metric spaces. However, our approach differs significantly, as it relies on a separation of scales and limits of discrete-time evolutions.

It is now relevant to highlight another related line of research, initiated by the seminal works [12, 17, 18, 19] on the construction of rate-independent BV-evolutions. The key distinction between this framework and both our setting and the previously mentioned works [28, 2, 25, 24] lies in the presence of a 1-homogeneous term in the energy functional, which introduces a dissipation term into the dynamics. In their pioneering work [17], A. Mielke, R. Rossi, and G. Savar´e construct evolutions as limits of viscous regularizations of solutions. Using similar techniques as in [12], they introduce the concept of parametrized metric solutions for rate-independent systems, which are absolutely continuous mappings from a parameter interval into an extended state space. In this framework, jumps are interpreted as generalized gradient flows, during which time remains constant, ultimately leading to BV-solutions. Notably, their formulation is developed entirely within the setting of metric spaces. In the follow-up paper [18] the same authors have revisited and aimed to clarify the key features of balanced viscosity solutions. The work is conducted in a finite-dimensional setting but employs two distinct convex functionals on the derivative of the solution: a 1-homogeneous functional, which is the standard choice for rate-independent systems (cf. [20] and references therein), and a superlinear functional that introduces viscous regularization. This approach offers significantly greater generality compared to the metric framework, which imposes the use of the same norm for both the rate-independent term and the quadratic regularization. As a result, a more refined analysis is required to understand the behavior of solutions during jumps. In the concluding work [19] the authors extend their results to the infinite-dimensional setting (Banach spaces). In this line of research as well, the approach involves constructing solutions as limits of a vanishing viscosity setting, where two distinct time scales coexist: one governing the evolution of critical points and the other regulating transitions during jumps. We further refer to the recent contributions [22, 23] for a similar research plan involving vanishing inertia and viscosity, where inertial effects appear in the energetic characterization of jump points.

To clarify how the present work is collocated in relation to [17, 18, 19] and the very vast related literature on rate-independent systems and doubly nonlinear equations (which we purposely do not report here), we reiterate the key differences: 1. We do not consider energy functionals with 1-homogeneous terms, nor do we seek BV rate-independent solutions. 2. Our construction explicitly separates/decouples the time scales of critical point evolution and jump transitions. 3. Our formulation is applicable in metric spaces.

We conclude this review of related work by noting that the separation/decoupling of time scales through local-in-time transitions was first explored in [8]. In that model, the energy was fixed, while the driving force of the dynamics was introduced via a timedependent constraint. Additionally, the analysis was conducted in Euclidean spaces setting and only an energy inequality (1.4) was obtained.

1.2. Plan of the Paper. In Section 2, we collect the assumptions that are needed in the results proved in the paper.

In Section 3, we propose an axiomatic construction of discrete quasistatic evolutions. Namely, the main result of the section (Theorem 3.9) states that, if such piecewise constant trajectories are defined through a transition rule that complies with Axioms 1 to 3, then the discrete quasistatic evolutions converge pointwisely—up to the extraction of a subsequence—to a limiting quasistatic curve taking values in a quotient space of

[0, T] ^× ^X. What is missing at this level, though, is a characterization of the energy jumps in the discontinuity points of the trajectory in terms of a variational action.

In Section 4, we tackle this point by introducing the notion of transition rule compatible with an action (see Definition 2) and of action generated by curves (see Definition 3). Namely, we first show that Axiom 3' implies Axiom 3, and that we can relate energy jumps to the values of the action (see Lemma 4.2). Then, we prove that, if a transition rule is compatible with an action generated by curves, then it satisfies Axiom 3' (see Lemma 4.8). Finally, we are in position for proving the main result of the paper (see Theorem 1 (complete)).

In Section 5 we show that transition rules obtained using gradient flow, minimizing movements and the BDF2 scheme are compatible with actions generated by curves (see Sections 5.1 to 5.3).

Finally, in Section 6 we present simple numerical experiments to illustrate our theoretical findings, where we simulate the breaking of an elastic rod.

2. General Notations and Assumptions

In this section, we collect the assumptions that are required throughout the paper. In each statement we shall explicitly list the needed assumptions. To ease the reader, we adopt similar notations as in [2]. We set our analysis on a metric space (X, d). We first state the basic structure that we need on (X, d).

Assumption 1. The space (X, d) is a locally compact metric space.

We make the following assumptions on the time-varying energy ^E : [0, T] ^× ^X ^→ ^R that drives the evolution.

Assumption 2. The function ^E : [0, T] ^× ^X ^→ ^R is continuous. Moreover, it is differentiable in the first variable, for every ^t ^∈ [0, T] and for every ^x ^∈ ^X, and the derivative ∂tE· : [0, T] ^× ^X ^→ ^R is continuous. Finally, for every ^t ^∈ [0, T] and ^x ^∈ ^X we consider the slope

$$|\partial Et|(x) := \limsup{x' \to x} \frac{\left(E_t(x) - E_t(x')\right)^+}{d(x, x')},$$

and we assume that |∂E· ^|: [0, T] ^× ^X ^→ ^R is a continuous function.

Remark 1. Under the continuity assumption and using local compactness, the slope coincides with the local Lipschitz constant

$$\limsup_{x'\to x} \frac{\left|E_t(x) - E_t(x')\right|}{d(x,x')}.$$

As the local Lipschitz constant is a strong upper gradient (as defined in [7, Def. 1.2.1]), under the preceding assumption, the slope is as well.

Assumption 3. There exist positive constants C1, C² > 0 such that

$$|\partial_t E_t(x)| \le C_1 E_t(x) + C_2$$

for every (t, x) ^∈ [0, T] ^× ^X.

From Assumption 3 it follows that ^Et(x) ≥ −C2/C¹ for every (t, x) ^∈ [0, T] ^× ^X, i.e., ^E : [0, T] ^× ^X ^→ ^R is bounded from below. Without loss of generality, we will assume throughout the paper that E is non-negative.

Assumption 4. We set ^G: ^X ^→ ^R as ^G(x) := supt∈[0,T] ^|Et(x)^| for every ^x ^∈ ^X, and we require that ^G is coercive. Namely, the sublevel sets {^x ^∈ ^X ^| ^G(x) ^≤ ^C} are compact in ^X for every ^C ^∈ ^R.

Remark 2. From Assumptions 2 to 4 it follows that E^t : ^X ^→ ^R is uniformly coercive in the time variable. Indeed, if we consider the set {^x ^∈ ^X ^| ^Et(x) ^≤ ^C} for ^C ^≥ 0, then from Assumption 3 and the Gr¨onwall Lemma we get Es(x) ≤ (C + C2|s − t|)e C1|s−t| for every ^s ^∈ [0, T] and for every ^x ^∈ ^X satisfying ^Et(x) ^≤ ^C. Hence, for every ^t ^∈ [0, T] we deduce the inclusion

$$\left{x \in \mathbb{X} \mid E_t(x) \le C\right} \subset \left{x \in \mathbb{X} \mid G(x) \le (C + C_2 T)e^{C_1 T}\right},\,$$

where the larger set (which does not depend on t) is compact by Assumption 4.

Assumption 5. For every t ∈ [0, T], the set C^t := {^x ^∈ ^X | |∂E^t |(x) = 0} can be expressed as the disjoint union of well-separated compacts, each of them being pathconnected with rectifiable paths. Moreover, for every path-connected component C ′ t ⊂ Ct , E^t : ^X ^→ ^R is constant on ^C ′ t .

Remark 3. We recall that in general the constancy of E^t on the connected components of C^t may fail (see [27]). However, this property is implied, e.g., by the fact that any two points in a connected component can be joined by a path ^η ^∈ AC([0, 1], ^X) (see [7, Section 1.1] for this notion) such that |∂E^t | η(s) = 0 for every s ∈ [0, 1]. Moreover, also from Assumption 6 below, it follows that E^t is constant on the path-connected components of C^t (see Remark 5).

Assumption 6. We require that, for every ^t ^∈ [0, T] and for every ^x ^∈ ^X with ^|∂E^t |(x) = 0 there exists a neighborhood U ∋ x and a function ε^x : [0, +∞) → [0, +∞) such that

$$E_t(x') - E_t(x) \ge -\varepsilon_x (d(x, x')) |\partial E_t|(x')$$ (2.1)

for every x ′ ∈ U, where εx(0) = lims→0⁺ εx(s) = 0.

Moreover, we assume that for a fixed ^u ^∈ ^X, the function ^t 7→ |∂E^t |(x) is Lipschitz continuous on [0, T], locally uniformly w.r.t. x.

Remark 4. We report that the inequality (2.1) in Assumption 6 is a reformulation of the more classical condition

$$\liminf_{x' \to x} \frac{E_t(x') - E_t(x)}{|\partial E_t|(x')} \ge 0$$

required for those x such that |∂E^t |(x) = 0 (see, e.g., [2, Assumption E4] and[25, Assumption F4]), under the hypothesis of isolated critical points. In our setting, the ratio written above is not suitable, as it is not defined whenever x lies in the interior of the set where |∂E^t | vanishes.

Remark 5. From Assumptions 2 and 6 it follows that for every path-connected component C ′ ^t ⊂ C^t , E^t : ^X ^→ ^R is constant on ^C ′ t . To see this, we show that, for every ^c ^∈ ^R, we have that both A^c := {x ∈ C ′ t ^| ^Et(x) ^≥ ^c} and ^A¯ c := {x ∈ C ′ t | Et(x) < c} are open in the topology induced by (X, d) on C ′ t . On the one hand, let x ∈ Ac, then by virtue of Assumption 6, there exists an open ball Bρ(x) such that Et(x ′ ) ≥ Et(x) ≥ c for every x ′ ∈ C ′ ^t ∩ Bρ(x) (here we combined Equation (2.1) with the fact that x ′ is a critical point). On the other hand, if ^x ^∈ ^A¯ ^c, then by continuity of E^t we deduce the existence of an open ball Bρ(x) such that Et(x ′ ) < c for every x ′ ∈ C ′ ^t ∩ Bρ(x). Since C ′ ^t ⁼ ^A^c ^∪ ^A¯ ^c for every ^c ^∈ ^R, we deduce that either ^C ′ ^t ⁼ ^A^c and ^A¯ ^c = ∅, or C ^t ⁼ ^A¯ c and A^c = ∅. Therefore, we have that, for every x ∈ C ′ t , ^Et(x) = sup{^c ^∈ ^R ^| ^A^c ̸⁼ ∅}.

It turns out that the time derivative ∂tE^t is constant on the connected components of critical points, for every but at most countably many exceptional instants t ∈ [0, T] (see Lemma 3.11). However, to show that, we need to strengthen the condition formulated in Equation (2.1).

Assumption 6'. We require that, for every ^t ^∈ [0, T] and for every ^x ^∈ ^X with |∂E^t |(x) = 0 there exists a neighborhood U ∋ x and a function ε^x : [0, +∞) → [0, +∞) such that

$$|E_t(x') - E_t(x)| \le \varepsilon_x (d(x, x')) |\partial E_t|(x')$$ (2.2)

for every x ′ ∈ U, where εx(0) = lims→0⁺ εx(s) = 0.

Moreover, we assume that the function t 7→ |∂E^t |(x) is Lipschitz continuous on [0, T], locally uniformly with respect to ^x ^∈ ^X.

Remark 6. We observe that the inequality (2.2) and its one-sided version (2.1) hold if E^t complies with the local Polyak- Lojasiewicz condition and if |∂E^t | is continuous (as prescribed in Assumption 2).

The next hypothesis will be used in the last part of the paper.

Assumption 7. (X, d) is a path space, i.e.,

$$d(x,x') = \inf \left{ \int_0^1 |\dot{\eta}(s)| \, \mathrm{d}s \ | \ \eta \in AC([0,1],\mathbb{X}), \, \eta(0) = x, \, \eta(1) = x' \right}.$$

Moreover, we require that |∂E^t ^|: ^X ^→ ^R Lipschitz continuous, uniformly as ^t varies in [0, T], i.e., there exists L > 0 such that

$$|\partial E_t|(x) - |\partial E_t|(x')| \le L d(x, x')$$

for every ^t ^∈ [0, T] and every x, x′ ^∈ ^X.

Remark 7. In Sections 5.2 and 5.3, we describe discrete minimizing schemes with a step size τ . Whenever we work with Assumption 7, we will assume that τ ≤ 1 L , where L is the Lipschitz constant assumed in Assumption 7.

Finally, we need the following assumption on the time derivative of the driving energy for proving that the measure involved in the energy balance is purely atomic. Before proceeding, we introduce the quotient space X as follows:

$$\mathcal{X} := ([0, T] \times \mathbb{X}) / \sim, \tag{2.3}$$

where ^∼ is the equivalence relation on [0, T] ^× ^X given by

$$(t_1, x_1) \sim (t_2, x_2) \iff t_1 = t_2 \text{ and } x_1, x_2 \text{ belong to the same}$$

path-connected component of ${x \in \mathbb{X} \mid |\partial E_{t_1}|(x) = 0}.$ (2.4)

We equip the space ^X with the quotient topology, and we denote with ^q : [0, T]×^X → X the quotient map. We recall that the quotient topology on X is the strongest that makes the mapping q continuous. Finally, we use the notation [(t, x)] to describe the elements of X , i.e., the equivalence classes induced by the relation (2.4).

Assumption 8. For every [(s, x)] ∈ X we have that ∂tEs(x1) = ∂tEs(x2), for every (s, x1),(s, x2) ∈ X . Hence, we can define ^∂tE^ˆ : X → ^R as ^∂tE^ˆ [(s, x)] := ∂tEs(x) for every [(s, x)] ∈ X . Furthermore, we require that ^∂tE^ˆ is continuous.

Remark ⁸. In the case the set {^x ^∈ ^X | |∂E^t |(x) = 0} consists of isolated points for every t ∈ [0, T], then Assumption 8 is implied by Assumption 2, as the quotient space X is homeomorphic to [0, T] ^× ^X.

3. Discrete quasistatic evolutions: axiomatic construction

In this section, we detail the properties that a family of piecewise constant curves should satisfy to be employed to retrieve (through a limiting argument) the solutions of Equation (1.1). We begin by introducing the curves that we use in our construction.

Definition 1 (Discrete quasistatic evolution). Let us assume Assumptions 1 and 2. Here, we fix a family of mappings ¯ω^t : ^X ^→ ^X indexed by ^t ^∈ [0, T], which we call the transition rule. Given δ > 0 and an initial state ^x⁰ ^∈ ^X such that ^|∂E0|(x0) = 0, we construct a discrete quasistatic evolution η δ : [0, T] ^→ ^X as follows:

We define M := ⌊T /δ⌋ (or M := T /δ − 1, if T /δ is integer).
We set w⁰ = x0.
For i = 1, . . . , M, we assign wⁱ = ¯ωiδ(wi−1).
For i = 0, . . . , M, for t ∈ [iδ,(i + 1)δ) (or t ∈ [iδ, T] in case i = M), we set η δ (t) = wⁱ .

We also introduce

$$J^{\delta} := { i\delta \mid i = 1, \dots, M }. \tag{3.1}$$

We observe that in Definition 1 we have not specified any rule for deriving ¯ω^t . One of the main contributions of the present paper is to provide a list of axioms that such a scheme should satisfy to be used for the construction—through a limiting argument—of a balanced viscosity quasistatic solution (see [2] for this notion).

In our arguments, the identities on the energy balance shall play a pivotal role. Given δ > 0 and a discrete quasistatic evolution η δ : [0, T] ^→ ^X constructed according to Definition 1, we define the function E·(η δ ): [0, T] ^→ ^R as

$$t \mapsto \mathcal{E}_t(\eta^{\delta}) := E_t(\eta^{\delta}(t))$$ (3.2)

for every t ∈ [0, T]. We observe that, owing to Assumption 2, the following limits exist:

$$\mathcal{E}t^+(\eta^\delta) := \lim{\tau \to t^+} \mathcal{E}\tau(\eta^\delta) = \lim{\tau \to t^+} E_\tau(\eta^\delta(\tau)),$$

$$\mathcal{E}t^-(\eta^\delta) := \lim{\tau \to t^-} \mathcal{E}\tau(\eta^\delta) = \lim{\tau \to t^-} E_\tau(\eta^\delta(\tau))$$

for every t ∈ (0, T). Moreover, we define a priori E 0 (η δ ) := E0(η δ ) and E + T (η δ ) := E^T (η δ ). We observe that by construction E·(η δ ) is continuous from the right.

3.1. Energy balance. In this part, we state the first axiom and we establish an energy balance for the discrete quasistatic evolutions that fulfill it. Moreover, we show at which extent the energy balance is preserved when we let δ tend to 0.

Axiom 1. Let the family of mappings ¯ω^t : ^X ^→ ^X indexed by ^t ^∈ [0, T] be the transition rule as in Definition 1. Then, we require that for every ^x ^∈ ^X and for every ^t ^∈ [0, T], we have that E^t ω¯t(x) ≤ Et(x).

We define the non-negative Radon measure µ ^δ ∈ M([0, T]) as follows:

$$\mu^{\delta}(B) := \sum_{t \in J^{\delta} \cap B} \left[ \mathcal{E}_t^{-}(\eta^{\delta}) - \mathcal{E}_t^{+}(\eta^{\delta}) \right]$$ (3.3)

for every Borel set B ⊂ [0, T]. On one hand, owing to Axiom 1, we observe that

$$\mu^{\delta}({t}) = \mathcal{E}_t^-(\eta^{\delta}) - \mathcal{E}_t^+(\eta^{\delta}) \ge 0 \tag{3.4}$$

for every t ∈ J δ (i.e., t = iδ with i = 1, . . . , M). On the other hand, µ({t}) = 0 if t ̸∈ J δ . We are now in position of establishing the energy balance identity for discrete quasistatic evolutions.

Lemma 3.1. Let us assume Assumptions 1 and 2. Given δ > 0, let η δ : [0, T] ^→ ^X be a discrete quasistatic evolution constructed according to Definition 1. Then, if Axiom 1 is satisfied, for every 0 ≤ t¹ ≤ t² ≤ T the following identity holds:

$$\mathcal{E}_{t2}^+(\eta^{\delta}) - \mathcal{E}{t1}^-(\eta^{\delta}) = \int{t_1}^{t_2} \partialt E{\tau}(\eta^{\delta}(\tau)) d\tau - \mu^{\delta}([t_1, t_2]), \tag{3.5}$$

where the non-negative measure µ δ is defined as in Equation (3.3).

Proof. Owing Assumption 2 and recalling that η δ is by construction piecewise constant, it follows that the function t 7→ Et(η δ ) is of class C ¹ on the interval (i−1)δ, iδ for every i = 1, . . . , M (as well as on the very last piece Mδ, T ). With an algebraic manipulation, we write

$$\mathcal{E}{t{2}}^{+}(\eta^{\delta}) - \mathcal{E}{t{1}}^{-}(\eta^{\delta}) = \left(\mathcal{E}{t{1}}^{+}(\eta^{\delta}) - \mathcal{E}{t{1}}^{-}(\eta^{\delta})\right) + \left(\mathcal{E}{i{1}\delta}^{-}(\eta^{\delta}) - \mathcal{E}{t{1}}^{+}(\eta^{\delta})\right) \

\sum{j=1}^{\ell} \left(\mathcal{E}{i{j}\delta}^{+}(\eta^{\delta}) - \mathcal{E}{i{j}\delta}^{-}(\eta^{\delta})\right) + \sum{j=1}^{\ell-1} \left(\mathcal{E}{i{j+1}\delta}^{-}(\eta^{\delta}) - \mathcal{E}{i{j}\delta}^{+}(\eta^{\delta})\right) \
\left(\mathcal{E}{t{2}}^{-}(\eta^{\delta}) - \mathcal{E}{i{\ell}\delta}^{+}(\eta^{\delta})\right) + \left(\mathcal{E}{t{2}}^{+}(\eta^{\delta}) - \mathcal{E}{t{2}}^{-}(\eta^{\delta})\right), \tag{3.6}$$

where {i¹ < . . . < iℓ} = J ^δ ^∩ (t1, t2). Using the Fundamental Theorem of Calculus and the measure µ ^δ defined in Equation (3.3), we rephrase Equation (3.6) as

$$\begin{split} \mathcal{E}^{+}_{t2}(\eta^{\delta}) - \mathcal{E}^{-}{t_1}(\eta^{\delta}) &= -\mu^{\delta}({t1}) + \int{t_1}^{i_1\delta} \partialt E{\tau}\big(\eta^{\delta}(\tau)\big) \,\mathrm{d}\tau + \sum_{j=1}^{\ell} -\mu^{\delta}({ij\delta}) \ &+ \sum{j=1}^{\ell-1} \int_{ij\delta}^{i{i+1}\delta} \partialt E{\tau}\big(\eta^{\delta}(\tau)\big) \,\mathrm{d}\tau + \int{i{\ell}\delta}^{t_2} \partialt E{\tau}\big(\eta^{\delta}(\tau)\big) \,\mathrm{d}\tau - \mu^{\delta}({t_2}), \end{split}$$

which yields Equation (3.5). □

Remark 9. In some circumstances, it can be useful to employ slight variations of Equa tion (3.5). For example, when dealing with the difference of the right limits E + t2 (η δ ) − E + t1 (η δ ), we observe that

$$\mathcal{E}_{t2}^+(\eta^{\delta}) - \mathcal{E}{t1}^+(\eta^{\delta}) = \mathcal{E}{t2}^+(\eta^{\delta}) - \mathcal{E}{t_1}^-(\eta^{\delta}) + \mu^{\delta}({t_1}),$$

yielding

$$\mathcal{E}_{t2}^+(\eta^\delta) - \mathcal{E}{t1}^+(\eta^\delta) = \int{t_1}^{t_2} \partialt E\tau(\eta^\delta(\tau)) d\tau - \mu^\delta((t_1, t_2])$$

for every 0 ≤ t¹ ≤ t² ≤ T. Similarly, we also have

$$\mathcal{E}_{t2}^-(\eta^\delta) - \mathcal{E}{t1}^-(\eta^\delta) = \int{t_1}^{t_2} \partialt E\tau(\eta^\delta(\tau)) d\tau - \mu^\delta([t_1, t_2)),$$

and

$$\mathcal{E}_{t2}^-(\eta^\delta) - \mathcal{E}{t1}^+(\eta^\delta) = \int{t_1}^{t_2} \partialt E\tau(\eta^\delta(\tau)) d\tau - \mu^\delta((t_1, t_2)).$$

We are now interested in studying the family (E·(η ^δⁿ ))n≥¹ when ^δⁿ ^→ 0 as ⁿ → ∞, under the hypothesis that the discrete quasistatic evolutions (η ^δⁿ )ⁿ are constructed using the same initial state ^x⁰ ^∈ ^X. We begin by proving a boundedness result.

Lemma 3.2. Let us assume Assumptions 1 to 3. For every δ > 0, let η δ : [0, T] ^→ ^X be a discrete quasistatic evolution constructed according to Definition 1 and starting from ^x⁰ ^∈ ^X. Then, if Axiom 1 is satisfied, we have

$$\sup_{t \in [0,T]} \mathcal{E}_t(\eta^{\delta}) \le E_0(x_0)e^{C_1(T+1)} + C_2 \frac{e^{C_1(T+1)} - 1}{C_1},\tag{3.7}$$

where C1, C² > 0 are the constants that appear in Assumption 3. In particular, the estimate is independent of δ.

Proof. Let us consider t ∈ [0, δ). Since η δ is constant on this interval, we have that E·(η δ ) is C 1 in [0, δ). Then, by virtue of Assumption 3 we compute

$$\left| \frac{\mathrm{d}}{\mathrm{d}t} \mathcal{E}_t(\eta^{\delta}) \right| = \left| \partial_t E_t(x_0) \right| \le C_1 E_t(x_0) + C_2 = C_1 \mathcal{E}_t(\eta^{\delta}) + C_2, \tag{3.8}$$

which yields

$$\sup_{t \in [0,\delta)} \mathcal{E}_t(\eta^{\delta}) \le \left( \mathcal{E}_0(\eta^{\delta}) + \delta C_2 \right) e^{\delta C_1},\tag{3.9}$$

where we applied the Gr¨onwall inequality. Moreover, repeating the same argument on the interval [δ, 2δ), we deduce that

$$\sup_{t \in [\delta, 2\delta)} \mathcal{E}t(\eta^{\delta}) \le \left( \mathcal{E}{\delta}(\eta^{\delta}) + \delta C_2 \right) e^{\delta C_1}. \tag{3.10}$$

However, recalling Definition 1 and Equation (3.2), in virtue of Axiom 1, we have that

$$\sup_{t\in[0,\delta)} \mathcal{E}t(\eta^{\delta}) \ge \mathcal{E}{\delta}^-(\eta^{\delta}) \ge \mathcal{E}{\delta}^+(\eta^{\delta}) = \mathcal{E}{\delta}(\eta^{\delta}),$$

that, together with Equations (3.9) and (3.10), implies

$$\sup_{t \in [0,2\delta)} \mathcal{E}t(\eta^{\delta}) \le \left(\sup{t \in [0,\delta)} \mathcal{E}_t(\eta^{\delta}) + C_2\delta\right) e^{C_1\delta} \le \mathcal{E}_0(\eta^{\delta}) e^{2\delta C_1} + \delta C2 \sum{j=1}^2 e^{j\delta C_1}.$$

With an inductive argument, it follows that

$$\sup_{t \in [0,T]} \mathcal{E}_t(\eta^{\delta}) \le \mathcal{E}_0(\eta^{\delta}) e^{(M+1)\delta C_1} + \delta C2 \sum{j=1}^{M+1} e^{j\delta C_1},$$

where M is the integer defined in Definition 1. By simple algebraic manipulation and basic properties of the exponential functions, we conclude for (3.7). □

The boundedness of the energy along discrete quasistatic evolutions implies that the trajectories themselves are uniformly bounded, as we shall see below.

Lemma 3.3. Let us assume Assumptions 1 to 4. For every δ > 0, let η δ : [0, T] ^→ ^X be a discrete quasistatic evolution constructed according to Definition 1 and starting from ^x⁰ ^∈ ^X. If Axiom 1 is satisfied, there exists a compact set ^K ^⊂ ^X such that ^η δ (t) ∈ K for every t ∈ [0, T] and every δ > 0.

Proof. From Lemma 3.2 it follows that there exists K¹ > 0 independent of δ such that

$$\sup_{t \in [0,T]} \mathcal{E}t(\eta^{\delta}) = \sup{t \in [0,T]} E_t(\eta^{\delta}(t)) \le K_1,$$

yielding Et(η δ (t)) ≤ K¹ for every t ∈ [0, T]. Owing to Remark 2 and of Assumption 4, we deduce the thesis. □

In the next result we study the variation of the functions t 7→ Et(η δ ) as we vary the parameter δ.

Lemma 3.4. Let us assume Assumptions 1 to 4. For every δ > 0, let η δ : [0, T] ^→ ^X be a discrete quasistatic evolution constructed according to Definition 1 and starting from ^u⁰ ^∈ ^X. Let us consider the function ^E·(^η δ ): [0, T] ^→ ^R defined in Equation (3.2). Then, if Axiom 1 is satisfied, there exists K > 0 independent of δ such that

$$\sup{\mathcal{P}} \sum{j=1}^{m} |\mathcal{E}{s{j+1}}(\eta^{\delta}) - \mathcal{E}{s{j}}(\eta^{\delta})| \le K, \tag{3.11}$$

where the supremum is taken over the family of finite partitions P = {0 = s¹ < s² < . . . < s^m = T} of [0, T].

Proof. Let ^m ^∈ ^N and let ^P ⁼ {0 = ^s¹ < s² < . . . < s^m ⁼ ^T} be a partition of [0, T]. Recalling that E·(η δ ) is continuous from the right, for every j = 1, . . . , m − 1 we have that

$$\mathcal{E}{s{j+1}}(u^{\delta}) - \mathcal{E}{s{j}}(u^{\delta}) = \mathcal{E}{s{j+1}}^{+}(\eta^{\delta}) - \mathcal{E}{s{j}}^{-}(\eta^{\delta}) + \mathcal{E}{s{j}}^{-}(\eta^{\delta}) - \mathcal{E}{s{j}}^{+}(\eta^{\delta}) = \int{s{j}}^{s{j+1}} \partial{t} E{\tau}(\eta^{\delta}(\tau)) d\tau - \mu^{\delta}([s{j}, s{j+1}]) + \mu^{\delta}({s{j}}) = \int{s{j}}^{s{j+1}} \partial{t} E{\tau}(\eta^{\delta}(\tau)) d\tau - \mu^{\delta}((s{j}, s_{j+1}]),$$ (3.12)

for every s^j < sj+1. Since µ δ is a non-negative measure, we deduce that

$$\sum{j=1}^{m} |\mathcal{E}{s{j+1}}(\eta^{\delta}) - \mathcal{E}{s{j}}(\eta^{\delta})| \le \int{0}^{T} \left| \partial{t} E{\tau} \left( \eta^{\delta}(\tau) \right) \right| d\tau + \mu^{\delta}([0, T]).$$ (3.13)

We further observe that, in virtue of Assumption 3 and Lemma 3.2, we have

$$\left| \partialt E\tau \left( \eta^\delta(\tau) \right) \right| \le C1 \mathcal{E}\tau(\eta^\delta) + C_2 \le K_1 \tag{3.14}$$

for every τ ∈ [0, T], where K¹ is a constant that does not depend on δ. Moreover, from Lemma 3.1 it descends that

$$\mu^{\delta}([0,T]) = \mathcal{E}_0(\eta^{\delta}) - \mathcal{E}_T(\eta^{\delta}) + \int_0^T \partialt E\tau(\eta^{\delta}(\tau)) d\tau,$$

and, combining the last identity with Equation (3.14) and with Lemma 3.1, we obtain

$$\mu^{\delta}([0,T]) \le K_2,$$ (3.15)

where, once again, K² > 0 does not depend on δ. Finally, combining Equations (3.13) to (3.15), we deduce the bound in Equation (3.11). □

We are now in position for establishing a result analogue to [2, Proposition 5.2].

Proposition 3.5. Let us assume Assumptions 1 to 4. Given a non-negative decreasing sequence (δn)ⁿ such that δⁿ → 0 as n → ∞, let η ^δⁿ : [0, T] ^→ ^X be discrete quasistatic evolutions constructed according to Definition 1 and starting from ^x⁰ ^∈ ^X. Then, if Axiom 1 is satisfied, there exist a positive Radon measure µ¯ ∈ M([0, T]), and functions E ∈¯ BV ([0, T], ^R) and D ∈ ^L∞([0, T], ^R) such that, up to a subsequence, for ⁿ → ∞ we have

$$\mu^{\delta_n} \rightharpoonup^* \bar{\mu} \quad \text{in } \mathcal{M}([0,T]),$$

$$\lim_{n \to \infty} \mathcal{E}_t(\eta^{\delta_n}) = \bar{\mathcal{E}}_t \quad \text{for every } t \in [0,T],$$

$$\mathcal{D}^{\delta_n} \rightharpoonup^* \mathcal{D} \quad \text{in } L^{\infty}([0,T],\mathbb{R}),$$ (3.16)

where we introduced the notation ^t 7→ Dδⁿ (t) := ^∂tE^t η ^δⁿ (t) . Moreover, if we use ^E¯⁺ t , ^E¯⁻ t to denote, respectively, the right and the left limits of ^E¯ at the instant ^t ^∈ 0, T, then we have that

$$\bar{\mathcal{E}}_{t2}^+ - \bar{\mathcal{E}}{t1}^- = \int{t_1}^{t_2} \mathcal{D}(\tau) \,\mathrm{d}\tau - \bar{\mu}([t_1, t_2]) \tag{3.17}$$

for every 0 ≤ t¹ ≤ t² ≤ T.

Proof. For every n ≥ 1, we define F ^δⁿ : [0, T] ^→ ^R as

$$\mathcal{F}_t^{\delta_n} := \mathcal{E}_t(\eta^{\delta_n}) - \int_0^t \mathcal{D}^{\delta_n}(\tau) \,d\tau$$ (3.18)

for every t ∈ [0, T]. We observe that the functions F ^δⁿ are non-increasing. Indeed, given 0 ≤ t¹ ≤ t² ≤ T we have that

$$\mathcal{F}_{t_2}^{\deltan} - \mathcal{F}{t_1}^{\deltan} = \mathcal{E}{t_2}^+(\eta^{\deltan}) - \mathcal{E}{t_1}^+(\eta^{\deltan}) - \int{t_1}^{t_2} \mathcal{D}^{\delta_n}(\tau) d\tau = -\mu^{\delta_n}((t_1, t_2]) \le 0,$$

where we used Remark 9. Moreover, the sequence (F ^δⁿ )ⁿ is uniformly bounded, owing to Equation (3.14) and Lemma 3.2. By Helly's Selection Theorem, it follows that up to a subsequence, (F ^δⁿ )ⁿ is pointwise convergent to a non-increasing function ^F¯ at every point t ∈ [0, T]. In addition, using again the estimate in Equation (3.14), it follows that the sequence (Dδⁿ )ⁿ is pre-compact in the weak-^∗ topology of ^L∞, while the measures (µ ^δⁿ )ⁿ are pre-compact in the weak-^∗ topology of ^M([0, T]). These observations establish the first and the third convergences in (3.16), along a proper and not relabelled subsequence. Finally, by passing to the limit in Equation (3.18), we get the pointwise convergence of the energy stated in (3.16). To prove the energy balance for ^E¯, we argue as in [2]. Namely, given 0 < t¹ ≤ t² < T, we compute

$$\bar{\mu}([t{1}, t{2}]) = \lim{k \to \infty} \bar{\mu}((t{1} - 1/k, t{2} + 1/k)) \leq \lim{k \to \infty} \liminf{n \to \infty} \mu^{\delta{n}}((t{1} - 1/k, t{2} + 1/k))$$

$$\leq \lim{k \to \infty} \liminf{n \to \infty} \mu^{\delta{n}}((t{1} - 1/k, t_{2} + 1/k])$$

$$= \lim{k \to \infty} \left(\bar{\mathcal{E}}{t{1} - 1/k} - \bar{\mathcal{E}}{t{2} + 1/k} + \int{t{1} - 1/k}^{t{2} + 1/k} \mathcal{D}^{\delta_{n}}(\tau) d\tau\right)$$

$$= \bar{\mathcal{E}}{t{1}}^{-} - \bar{\mathcal{E}}{t{2}}^{+} + \int{t{1}}^{t_{2}} \mathcal{D}(\tau) d\tau.$$ (3.19)

Moreover, we have

$$\begin{split} \bar{\mu}([t_1,t2]) &= \lim{k \to \infty} \bar{\mu}([t_1 - 1/k, t2 + 1/k]) \geq \lim{k \to \infty} \limsup_{n \to \infty} \mu^{\delta_n}([t_1 - 1/k, t2 + 1/k]) \ &\geq \lim{k \to \infty} \limsup_{n \to \infty} \mu^{\delta_n}((t_1 - 1/k, t2 + 1/k]) \ &= \lim{k \to \infty} \left( \bar{\mathcal{E}}_{t1 - 1/k} - \bar{\mathcal{E}}{t2 + 1/k} + \int{t_1 - 1/k}^{t_2 + 1/k} \mathcal{D}^{\deltan}(\tau) \, \mathrm{d}\tau \right) \ &= \bar{\mathcal{E}}{t1}^- - \bar{\mathcal{E}}{t2}^+ + \int{t_1}^{t_2} \mathcal{D}(\tau) \, \mathrm{d}\tau, \end{split} \tag{3.20}$$

and this concludes the proof. □

In the next corollary we report some further properties of the limiting function ^E¯ constructed above.

Corollary 3.6. Under the same assumptions and notations as in Proposition 3.5, we deduce that the function ^E¯: [0, T] ^→ ^R is of bounded variation, that its distributional derivative satisfies ^dE¯ ⁼ DL¹ ⁻ ^µ¯, and that

$$\bar{\mathcal{E}}_t^+ - \bar{\mathcal{E}}_t^- = -\bar{\mu}({t})$$ for every $t \in [0, T]$ .

Finally, the discontinuity points of ^E¯ (i.e., the atoms of ^µ¯) are at most countably many.

Proof. The proof directly descends from (3.19) and (3.20). □

3.2. Limiting construction of quasistatic evolutions. In this part, we show how we can obtain quasistatic evolutions using a family of curves (η δ )^δ constructed according to Definition 1. We introduce below the second axiom.

Axiom 2. Let the family of mappings ¯ω^t : ^X ^→ ^X indexed by ^t ^∈ [0, T] be the transition rule as in Definition 1. Then, we require that for every ^x ^∈ ^X and for every ^t ^∈ [0, T], we have that |∂E^t | ω¯t(x) = 0.

In the next result, we show that the curves η ^δ are somehow close to be made of critical points, even for t ̸∈ J δ . More precisely, we provide an estimate uniform in δ for the magnitude of |∂E^t | along η δ

Lemma 3.7. Let us assume Assumptions 1 to 4. For every δ > 0, let η δ : [0, T] ^→ ^R d be a discrete quasistatic evolution constructed according to Definition 1 and starting from ^x⁰ ^∈ ^X. Then, if Axioms 1 and 2 are satisfied, for every ε > ⁰ there exists ¯δ > ⁰ such that for every δ ∈ (0, ¯δ]

$$\sup_{t\in[0,T]}|\partial E_t|\big(\eta^\delta(t)\big)\leq\varepsilon.$$

Proof. Recalling the definition of J δ in Equation (3.1), we observe that the set J ^δ∪{0} ⁼ {iδ | i = 0, . . . , M} is a δ-net for the interval [0, T]. In virtue of Lemma 3.3 and of Assumption 4, there exists a compact set ^K ^⊂ ^X such that ^η δ (t) ∈ K for every t ∈ [0, T] and for every δ > 0. Moreover, owing to Assumption 2, it descends that the space-gradient |∂E· ^|: [0, T] ^× ^X ^→ ^R is uniformly continuous when restricted to [0, T] ^× ^K, and we denote with ^ξ : [0, T] ^× [0, diam(K)] ^→ ^R⁺ a modulus of continuity. We recall that ξ is a function non-decreasing in each argument and that satisfies ξ(0, 0) = lim(s,r)→(0+,0+) ^ξ(s, r) = 0. Let us fix ^t ^∈ [0, T]. Then, there exists ^ˆⁱ ∈ {0, . . . , M} such that ^ˆiδ ^≤ t < (ˆⁱ + 1)δ. By Axiom 2 and recalling that ^η δ is piecewise constant, we observe that |∂Eˆiδ| η δ (t) = 0. Therefore, we have that

$$|\partial Et| \left( \eta^{\delta}(t) \right) \le |\partial E{\hat{t}\delta}| \left( \eta^{\delta}(t) \right) + \left| |\partial Et| \left( \eta^{\delta}(t) \right) - |\partial E{\hat{t}\delta}| \left( \eta^{\delta}(t) \right) \right| \le \xi(\delta, 0),$$

and this concludes the proof. □

Remark 10. A similar result is reported in [2, Notation 5.3] for the class of curves obtained by solving a properly rescaled gradient flow. However, in the construction in [2], it is possible to prove that the set of instants where the gradients converge to 0 has full Lebesgue measure, but not that it is the whole interval [0, T].

Before proceeding, we recall the definition of the quotient space X given in Equa tion (2.3):

$$\mathcal{X}\coloneqq \big([0,T]\times\mathbb{X}\big)/\sim,$$

where ^∼ is the equivalence relation on [0, T] ^× ^X given by (cf. Equation (2.4))

$$(t_1, x_1) \sim (t_2, x_2) \iff t_1 = t_2 \text{ and } x_1, x_2 \text{ belong to the same}$$

path-connected component of $${x \in \mathbb{X} \mid |\partial E_{t_1}|(x) = 0}$$ .

The space ^X is equipped with the quotient topology. We denote with ^q : [0, T] ^× ^X → X the quotient map, and we use the notation [(t, x)] to describe the elements of X , i.e., the equivalence classes induced by (2.4). We will often denote by ˆx an element [(t, x)] of X .

Lemma 3.8. Let us assume Assumptions 1 and 5. Then, the quotient space X defined in (2.3) with the relation (2.4) is a Hausdorff space, i.e., for every xˆ1, xˆ² ∈ X with xˆ¹ ̸= ˆx² there exist two disjoint open sets U1, U² ⊂ X such that xˆ¹ ∈ U¹ and xˆ² ∈ U2.

Proof. For every ˆ^x = [(t, x)] ∈ X , let us introduce the set ^A := {^w ^∈ ^X ^| (t, w) ^∼ (t, x)}, which satisfies either A = {x} in the case |∂E^t |(x) ̸= 0, or, if |∂E^t |(x) = 0, A does coincide with the path-connected component containing x of the critical points of E^t . Moreover, from Assumption 5 it descends that A is compact. Then, observing that q −1 (ˆx) = {t} × A, we deduce the thesis by applying [15, Theorem 8.11 and Exercise 8.13.(l)], and recalling that [0, T] ^× ^X is a Hausdorff space. □

We observe that Lemma 3.8 implies the uniqueness of the limit for every converging sequence in ^X . Furthermore, we notice that the energy ^E : [0, T]×^X ^→ ^R can be defined also on the quotient space X . Namely, given ˆx = [(t, x)] ∈ X , we set

$$\hat{E}(\hat{x}) = \hat{E}([(t,x)]) := E_t(x).$$ (3.21)

Let us show that this definition does not depend on the representative of the class. Indeed, if |∂E^t |(x) ̸= 0, we have that [(t, x)] = {(t, x)}. Otherwise, if |∂E^t |(x) = 0, let us consider another element of the same class (t, x′ ) ∈ (t, x), and we observe that Et(x) = Et(x ′ ) by virtue of Assumption 5. This shows that ^E^ˆ : X → ^R is well-defined, and we have that ^E ⁼ ^E^ˆ ◦ ^q. The last identity (together with the continuity of ^E) implies that ^E^ˆ : X → ^R is continuous as well (see the universal property of quotients [15, Theorem 5.2]).

In the next fundamental axiom, we state a property for the limits of the sequences η ^δⁿ (t n 1 ) n , η ^δⁿ (t n 2 ) n when t n 1 , tⁿ ² → t as n → ∞. In the limiting construction, this axiom plays the same role as [2, Lemma 5.1].

Axiom 3. Given a non-negative decreasing sequence (δn)ⁿ such that δⁿ → 0 as n → ∞, let η ^δⁿ : [0, T] ^→ ^X be discrete quasistatic evolutions constructed according to Def inition 1 and starting from ^x⁰ ^∈ ^X. Let us further assume that along η δn n the convergences reported in (3.16) hold. For every t ∈ [0, T], let us consider sequences (t n 1 )n,(t n 2 )ⁿ ^⊂ [0, T] and ^x1, x² ^∈ ^X such that ^t n 1 , tⁿ ² → t as n → ∞ with t n ¹ ≤ t n 2 for every n, and such that η ^δⁿ (t n 1 ) → x¹ and η ^δⁿ (t n 2 ) → x² as n → ∞. If x¹ and x² belong to different path-connected components of the set {^x ^∈ ^X | |∂E^t |(x) = 0}, then there exists c > 0 such that

$$\bar{\mu}({t}) \ge c, \qquad E_t(x_1) - E_t(x_2) \ge c.$$

Remark 11. If compared to Axioms 1 and 2, we notice that Axiom 3 sounds intrinsically different and less elegant, as it does not directly involve the transition rules (¯ωt)t∈[0,T] , but it is rather formulated in terms of the subsequence constructed in Proposition 3.5. We shall devote Section 4 to amend this point: We formulate Axiom 3', we show that Axioms 1, 2 and 3' imply Axiom 3 (see Lemma 4.2), and finally in Definition 3 we describe a class of transition rules that comply with Axiom 3' (see Lemma 4.8). Examples of such transition rules are provided in Section 4.2.

Remark 12. Using the same notations as in Axiom 3, the points x1, x² are automatically critical points. Indeed, from the construction of η ^δⁿ and from Lemma 3.7, it follows that

$$\lim{n \to \infty} \sup{s \in [0,T]} |\partial E_s| (\eta^{\delta_n}(s)) = 0.$$

Hence, from the continuity of |∂E· ^|: [0, T] ^× ^X ^→ ^R, we conclude that ^|∂E^t |(x1) = 0. The argument for x² is exactly the same.

Remark 13. The previous axiom is particularly meaningful for the instants t ∈ [0, T] such that ¯µ({t}) = 0. Indeed, in this case we conclude that the points ^x1, x² ^∈ ^X obtained as above must belong to the same path-connected component of {^x ^∈ ^X | |∂E^t |(x) = 0}. In particular, in case that the critical points of E^t are isolated (as assumed in [2]), it descends that x¹ = x2.

Remark 14. Under Axiom 1, it is possible to prove a sort of converse of Axiom 3. Namely, using the same notations, let us assume that along η δn n the convergences reported in (3.16) hold, and that there exists t ∈ (0, T) such that ¯µ({t}) = c > 0. Then, following the arguments of [2, Proposition 4.1] it is possible to construct a subsequence η δnk k and sequences (t k 1 )k,(t k 2 )^k satisfying t k ¹ ≤ t k 2 and t k 1 , t^k ² → t as k → ∞, such that u ^δnk (t k 1 ) → x¹ and u ^δnk (t k 2 ) → x2, where x1, x² belongs to different connected components of the set of critical points of E^t .

We are finally in position to construct a solution ˆη : [0, T] → X obtained as the pointwise limit of (the graphs of) discrete quasistatic evolutions. Our construction of ˆη follows steps similar to the ones detailed in [2] for their framework.

Theorem 3.9. Let us assume Assumptions 1 to 5. Given a non-negative decreasing sequence (δn)ⁿ such that δⁿ → 0 as n → ∞, let η ^δⁿ : [0, T] ^→ ^X be discrete quasistatic evolutions constructed according to Definition 1 and starting from ^x⁰ ^∈ ^X. Let us suppose that the convergences (3.16) hold and that Axioms 1 to 3 are satisfied. Then, there exists a curve ηˆ: [0, T] → X such that

The right limit ηˆ ⁺(t) exists for every ^t ^∈ (0, T] and left limit ^η^ˆ −(t) exists for every t ∈ (0, T].
For every ^t ^∈ [0, T], ^ηˆ(t) ⊂ {(t, x) ^∈ [0, T] ^× ^X | |∂E^t |(x) = 0}.
For every 0 ≤ s ≤ t ≤ T, the following energy balance holds:

$$\hat{E}(\hat{\eta}^+(t)) - \hat{E}(\hat{\eta}^-(s)) = \int_s^t \mathcal{D}(\tau) d\tau - \bar{\mu}([s,t]), \qquad (3.22)$$

where D ∈ ^L∞([0, T, ^R) and ^µ¯ ∈ M+([0, T]) are defined in Proposition 3.5. In particular, the set J of atoms of µ¯ is at most countable.

ηˆ is continuous in [0, T] \ J, J coincides with the jump set of ηˆ, and for every ^t ^∈ [0, T] we have ^E^ˆ ηˆ −(t) ⁻ ^E^ˆ ηˆ ⁺(t) = ¯µ({t}).
There exists a subsequence (δn^k )^k such that t, ηδnk (t) →^X ηˆ(t) for all t ∈ [0, T].

Proof. Step 1: Definition of the limiting trajectory on a countable dense set. Let us consider the set I := J ∪ A, where J := {t ∈ [0, T] | µ¯({t}) > 0} is the (at most) countable set of discontinuity points of ^E¯ , and A is a countable set dense in [0, T]. Then, since by Lemma 3.3 there exists ^K ^⊂ ^X compact such that ^η ^δⁿ (t) ^∈ ^K for every t ∈ [0, T] and every n ≥ 1, we may construct a subsequence (η ^δⁿ )ⁿ that is convergent at every ^t ^∈ ^I. Therefore, we define ^ηtemp : ^I ^→ ^X as ^ηtemp(t) := limn→∞ ^η ^δⁿ (t). In order to manage the connected components of the critical points of the driving energy, it is convenient to introduce the curve function ˆη : I → X , defined as the composition ηˆ(t) := q (t, ηtemp(t)) = [(t, ηtemp(t))].

Step 2: Extension of ηˆ to the whole evolution interval. We show that ˆη admits a unique extension at any point t ∈ [0, T] \ I. First of all, let us consider a sequence (t k ) ⊂ I such that t^k → t ̸∈ I as k → ∞. We want to show that the sequence ηˆ(t k ) k = [(t k , ηtemp(t k ))] k ⊂ X admits a converging subsequence in X . Since ηtemp takes value in the compact set ^K ^⊂ ^X, we can extract a (not relabelled) subsequence such that (t k , ηtemp(t k )) k converges to (t, x) in [0, T] ^× ^X. Therefore, by the continuity of ^q : [0, T] ^× ^X → X , along such a subsequence we have that (ˆη(^t k ))^k = q t k , ηtemp(t k ) k converges in X to [(t, x)] = q((t, x)). Hence, we can set ˆη(t) := [(t, x)] for t ̸∈ I. We have now to show that such an extension is uniquely defined.

To see this, let us assume that there exist (t k 1 )k,(t k 2 )^k ⊂ I such that limk→∞ t k ¹ = limk→∞ t k ² = t ̸∈ I, and let us assume that ηˆ(t k 1 ) k , ηˆ(t k 2 )) k ⊂ X have limits in X . We shall prove that the two limits coincide by considering for every k the elements t k 1 , ηtemp(t k 1 ) ∈ ηˆ(t k 1 ) and t k 2 , ηtemp(t k 2 ) ∈ ηˆ(t k 2 ). Arguing as above, up to a not relabelled subsequence, we may assume that t k 1 , ηtemp(t k 1 ) k and t k 1 , ηtemp(t k 1 ) k converge in [0, T] ^× ^X to (t, x1),(t, x2), respectively. The extension is unique if we show that (t, x1) ∼ (t, x2).

Since ηtemp is defined as the pointwise limit of (η ^δⁿ )ⁿ on I, with a diagonal procedure we can extract a subsequence n^k such that x¹ = limk→∞ η ^δnk (t k 1 ) and x² = limk→∞ η ^δnk (t k 2 ). We may further assume that t k ¹ ≤ t k 2 or t k ¹ ≥ t k 2 for every k ≥ 1. Since t ̸∈ I and J ⊂ I, we have that ¯µ({t}) = 0. Therefore, in virtue of Axiom 3, we deduce that x¹ and x² belong to the same connected component of the critical points of E^t , i.e., (t, x1) ∼ (t, x2) according to Equation (2.4).

Hence, we can uniquely extend ˆη to [0, T].

Step 3: Pointwise convergence on the whole evolution interval. Let (η ^δⁿ )ⁿ be the sequence of discrete quasistatic solutions that converge pointwisely to ˆηtemp on I. For t ̸∈ I, we consider the sequence of points η ^δⁿ (t) n ⊂ K. We aim to show that, if x ′ ^∈ ^X is a limiting point of η ^δⁿ (t) n , then [(t, x′ )] = ˆη(t), so that we can conclude that t, ηδⁿ (t) ] →^X ηˆ(t) in X as n → ∞. To see that, let us restrict to a (not relabelled) subsequence such that x ′ = limn→∞ η ^δⁿ (t). Moreover, let us take a sequence of instants I ∋ t ^k ↗ ^t as ^k → ∞ and such that ^x = limk→∞ ^ηtemp(^t k ), and, by virtue of Step 2, we have that ˆη(t) = [(t, x)] and x = limk→∞ η ^δnk (t k ), where the last limit is computed along a suitable subsequence of (η ^δⁿ )ⁿ obtained with a diagonal procedure. If we use Axiom 3 on the sequences η ^δnk (t) k and η ^δnk (t k ) k , from the fact that ¯µ({t}) = 0 (we recall that t ̸∈ I), we deduce that x and x ′ must lie in the same connected component of C^t , i.e., [(t, x′ )] = [(t, x)] = ˆη(t).

Hence, we deduce that the sequence (η ^δⁿ )n, which converges to ηtemp pointwise on I, satisfies as well [ t, ηδⁿ (t) ] →^X ηˆ(t) in X as n → ∞ for every t ∈ [0, T].

Step 4: Driving energy along ηˆ. Owing to the convergences (3.16) in Proposition 3.5, we have that

$$\bar{\mathcal{E}}t = \lim{n \to \infty} \mathcal{E}_t(\eta^{\deltan}) = \lim{n \to \infty} E_t(\eta^{\delta_n}(t)) = \hat{E}(\hat{\eta}(t)), \tag{3.23}$$

where we used Step 3 in the last identity.

Step 5: The limiting trajectory ηˆ admits left and right limits. Assume the sequences (t k 1 )k,(t k 2 )^k are such that t k ¹ ↘ t, t k ² ↘ t as k → ∞, and ˆx¹ := limk→∞ ηˆ(t k 1 ) and ˆx² := limk→∞ ηˆ(t k 2 ). Without loss of generality, we may assume that for every k ≥ 1 we have t k ¹ ≤ t k 2 or t k ¹ ≥ t k 2 . If (η ^δⁿ )ⁿ is the sequence of discrete quasistatic evolutions

constructed in Step 1, we recall that, by Lemma 3.3, for every t ∈ [0, T] the sequence η ^δⁿ (t) n is contained in the compact subset ^K ^⊂ ^X. Moreover, if ^y ^∈ ^X is a limiting point of η ^δⁿ (t) n , from Step 3 it follows that (t, y) ∈ ηˆ(t). Let us consider the sets η ^δⁿ (t k 1 ) n,k, η ^δⁿ (t k 2 ) n,k indexed by n, k ^∈ ^N, and, with a diagonal argument on the index n, we extract a (not relabelled) subsequence in n such that η ^δⁿ (t k 1 ) → y k 1 and η ^δⁿ (t k ) → y k 2 in ^X as ⁿ → ∞. As recalled above, we have that (^t k 1 , y^k 1 ) ∈ ηˆ(t k 1 ) and (t k 2 , y^k 2 ) ∈ ηˆ(t k 2 ) for every ^k ^∈ ^N. Moreover, up to the extraction of a subsequence in ^k, we may assume that y k ¹ → y¹ and y k ² ^→ ^y² in ^X as ^k → ∞, and we get that (t, y1) ^∈ ^xˆ¹ and (t, y2) ∈ xˆ2. Therefore, to show that ˆx¹ = ˆx² (i.e., that the limits of ˆη computed along (t k 1 )k,(t k 2 )^k coincide) it suffices to prove that (t, y1) ∼ (t, y2).

To see that, we first establish the following identities on the energy:

$$\bar{\mathcal{E}}{t}^{+} = \lim{\tau \to t^{+}} \bar{\mathcal{E}}{\tau} = \lim{k \to \infty} \bar{\mathcal{E}}{t{1}^{+}} = \lim{k \to \infty} \hat{E}(\hat{\eta}(t{1}^{k})) = \hat{E}(\hat{x}{1}) = E{t}(y{1}) \bar{\mathcal{E}}{t}^{+} = \lim{\tau \to t^{+}} \bar{\mathcal{E}}{\tau} = \lim{k \to \infty} \bar{\mathcal{E}}{t{2}^{k}} = \lim{k \to \infty} \hat{E}(\hat{\eta}(t{2}^{k})) = \hat{E}(\hat{x}{2}) = E{t}(y{2}).$$ (3.24)

Now, for every k ≥ 1, we construct n^k > nk−¹ such that

$$\max\left{|\eta^{\delta_{n_k}}(t_1^k)-y1^k|,|\eta^{\delta{n_k}}(t_2^k)-y_2^k|\right}\leq \frac{1}{k},$$

so that we obtain η ^δnk (t k 1 ) → y¹ and η ^δnk (t k 2 ) ^→ ^y² in ^X as ^k → ∞. Invoking Axiom 3, from Equation (3.24) we deduce that y¹ and y² belong to the same path-connected component of C^t , i.e., (t, y1) ∼ (t, y2). Moreover, since the curve τ 7→ ηˆ(τ ) in X admits limit from the right, we further deduce that

$$\bar{\mathcal{E}}t^+ = \lim{\tau \to t^+} \hat{E}(\hat{\eta}(\tau)) \tag{3.25}$$

for every t ∈ [0, T), and we set ˆη ⁺(t) := limτ→t⁺ ^ηˆ(^τ ). The same arguments and conclusions hold for the left limits. Finally, if ¯µ({t}) = 0 for t ∈ (0, t), we obtain that ˆη is continuous at t.

Step 6: Energy balance with the limiting trajectory. From Equations (3.17) and (3.25), recalling that ^E^ˆ : X → ^R is continuous, we deduce Equation (3.22). In particular, we obtain that Eˆ ηˆ ⁻(t)) ⁻ ^E^ˆ ηˆ ⁺(t) = ¯µ({t}) for every ^t ^∈ [0, T]. □

Remark 15. As the curve ˆη : [0, T] → X is obtained as point-wise limit of discrete quasistatic curves η ^δⁿ such that η ^δⁿ (t) ^∈ ^K for every ^t ^∈ [0, T] and every ⁿ ^≥ 1, for some compact subset K of X, it turns out that q −1 (ˆη) ⊂ [0, T] × K. Therefore, if we consider any sequence of representatives (tⁱ , xi) ^∈ ^ηˆ(ti) for ⁱ ^∈ ^N, it turns out that the sequence (xi)i∈^N ^⊂ ^X is pre-compact, owing to Assumption 1.

Let us stress that at this point, we have achieved the existence of a limit curve ˆη with an energy balance (see Equation (3.22)) and where Eˆ ηˆ −(t) ⁻ ^E^ˆ ηˆ ⁺(t) = ¯µ({t}) on the jump set J. Comparing this to the energy balance obtained in [2, Eq. 1.9a and 1.9b] within a more restrictive set of assumptions, there are still two refinement steps to be taken:

(1) The function D which appears in the identity in (3.22) is defined in Proposi tion 3.5 as the weak-∗ limit ∂tE·(η ^δⁿ (·)) ⇀^∗ ^L^∞ D as n → ∞, but we have not yet related D directly to the limit curve ˆη. In contrast, in [2, Eq. 1.9a], inside the integral term of the energy balance we read the evaluation of ∂tE· on the limit curve. To close this gap, we would like to relate D(t) to ∂tE^t and to ˆη(t).

(2) Our energy balance does not yet ensure that ˆη does not cross energy barriers. In contrast, in [2, Eq. 1.9b], the authors show that

$$\hat{E}(\hat{\eta}^{-}(t)) - \hat{E}(\hat{\eta}^{+}(t)) = \bar{\mu}({t}) = c_{t}(\hat{\eta}^{-}(t), \hat{\eta}^{+}(t)), \tag{3.26}$$

where c^t : R ^d ^× ^R ^d ^→ ^R is a cost function defined by minimizing the energydissipation integrals; Equation (3.26) prevents the limit trajectory form traversing energy barriers. Likewise, we would like to establish a similar relation for our energy balance.

In the rest of this section and the next two, we address the points raised above: In Section 3.2.1, we work towards Corollary 3.12, which shows that D(t) ≡ ∂tEt(ˆη(t))—in a suitable sense, as ˆη takes values in X—if E fulfills Assumption 6. In Section 4 and Section 5, we investigate for which transition rules ¯ω^t we can find a characterization of the jumps of ˆη analogous to (3.26), and thus rule out that ˆη crosses energy barriers.

3.2.1. Relating D(t) to ∂tE·. When trying to relate D(t) to ∂tE^t and to ˆη(t), we run into the limitation that ˆη takes values in the quotient space X , while the ∂tE· a priori does not factor through X—and thus, the quantity ∂tEt(ˆη(t)) is not even well-defined. We will see, however, that ∂tE· factors through X for almost every t ∈ [0, T] if E fulfills Assumption 6. To start, let us first provide upper and lower bounds for D—we will see later that those lower and upper bounds coincide for almost all t ∈ [0, T].

Lemma 3.10. Let us assume that the requirements of Theorem 3.9 are fulfilled and let ηˆ be a limit curve constructed according to Theorem 3.9. Then

$$\sup_{(t,x)\in\hat{\eta}(t)} \partial_t Et(x) \ge \mathcal{D}(t) \ge \inf{(t,x)\in\hat{\eta}(t)} \partial_t E_t(x)$$ (3.27)

for a.e. t ∈ [0, T].

Proof. Recalling the pointwise convergence established in Step 3 in the proof of Theo rem 3.9, for every t we have that, given a subsequence η ^δnk (t) k such that η ^δnk (t) ^→ ^x¹ as k → ∞, then (t, x1) ∈ ηˆ(t) and

$$\lim{k \to \infty} \mathcal{D}^{\delta{nk}}(t) = \lim{k \to \infty} \partial_t Et(\eta^{\delta{n_k}}(t)) = \partial_t E_t(x1) \ge \inf{(t,x) \in \hat{\eta}(t)} \partial_t E(x). \tag{3.28}$$

Likewise, we get that limk→∞ D ^δnk (t) ^≤ sup(t,x)∈ηˆ(t) ^∂tE(x). Furthermore, since by Lemma 3.3 there exists ρ > 0 such that η ^δⁿ (t) ^∈ ^Bρ(x0), we have that for every ^t ^∈ [0, T], ∂tE^t η ^δⁿ (t) is uniformly bounded. Thus, from Fatou's Lemma we deduce that

$$\limsup_{n\to\infty} \mathcal{D}^{\deltan}(t) \ge \mathcal{D}(t) \ge \liminf{n\to\infty} \mathcal{D}^{\delta_n}(t)$$

for a.e. t ∈ [0, T]. Since we can approximate the lim sup with a subsequence, by combining the last inequality with Equation (3.28) we conclude the proof. □

In the next result, we show that, if we strengthen the inequality (2.1) in Assump tion 6 by making it symmetric (see Assumption 6'), the infimum and the supremum in Lemma 3.10 coincide almost everywhere.

Lemma 3.11. Let us assume Assumptions 1 to 5 and 6'. Let ^K ^⊆ ^X be compact. Then for all but countable t ∈ [0, T], and for all connected components {C t i }i∈^I of {^x ^∈ ^X : ^|∂Et(x)^| = 0} ∩ ^K, there exists some ^κⁱ ^∈ ^R such that ^∂tEt(x) = ^κⁱ for every ^x ^∈ ^C t i

Before proving Lemma 3.11, let us note that Lemma 3.11 together with Lemma 3.10 fully characterize D almost everywhere. To this end, Lemma 3.2 and Assumption 4 ensure that we can find a compact K as required in Lemma 3.11.

Corollary 3.12. Let us assume Assumptions 1 to 5 and 6', and furthermore that the requirements of Theorem 3.9 are fulfilled. Let ηˆ: [0, T] → X be a curve constructed according to Theorem 3.9. Then, we have

$$\mathcal{D}(t) = \inf_{(t,x)\in\hat{\eta}(t)} \partial_t Et(x) = \sup{(t,x)\in\hat{\eta}(t)} \partial_t E_t(x)$$ (3.29)

for a.e. t ∈ [0, T]. In particular, we can rewrite the energy balance Equation (3.22) as

$$\hat{E}(\hat{\eta}^{+}(t)) - \hat{E}(\hat{\eta}^{-}(s)) + \bar{\mu}([t{1}, t{2}]) = \int{t{1}}^{t{2}} \mathfrak{D}{t}^{+} E{\tau}(\hat{\eta}(\tau)) d\tau = \int{t{1}}^{t{2}} \mathfrak{D}{t}^{-} E{\tau}(\hat{\eta}(\tau)) d\tau,$$

where D + ^t E^τ (ˆη(τ )) := sup(τ,x)∈ηˆ(τ) ∂tE^τ (x) and D − ^t E^τ (ˆη(τ )) := inf (τ,x)∈ηˆ(τ) ∂tE^τ (x).

Proof of Lemma 3.11. We set ^C^t to be the set of connected components of {^x ^∈ ^X : |∂E^t |(x) = 0} ∩ K. Defining

$$H(t) = \sup_{\mathscr{C} \in \mathfrak{C}t} \sup{x,y \in \mathscr{C}} \left( \partial_t E_t(x) - \partial_t E_t(y) \right),$$

we thus need to show that for all but countable t, H(t) = 0. Since we have {t ∈ [0, T] : H(t) > 0} = S ⁿ∈^N {t ∈ [0, T] : H(t) > 1/n}, it suffices to show that for each ε > 0, we have that H(t) ≤ ε for all but countable t. We will proceed by contradiction and assume that there is some ε > 0 which does not satisfy such condition, i.e., B^ε := {t ∈ [0, T] : H(t) > ε} is not at most countable. Within Bε, we can choose an increasing sequence which converges to ^t¯ := inf{^t ^∈ [0, T] ^| ^B^ε ^∩ [t, T] is countable}. Thus, we can find a sequence of triples (tn, xn, yn) ^∈ [0, T] ^× ^X ^× ^X such that

(1) ^tⁿ ^↑ ^t¯
(2) ^tⁿ ̸⁼ ^t¯
(3) |∂Etⁿ |(xn) = |∂Etⁿ |(yn) = 0
(4) Etⁿ (xn) = Etⁿ (yn) (by Assumption 5)
(5) ∂tEtⁿ (xn) > ∂tEtⁿ (yn) + ε
(6) xⁿ ∈ K, yⁿ ∈ K.

Because of the last condition, we can—without relabeling—extract a subsequence such that, in addition to the conditions above, it holds that (xn, yn) ^→ (¯x, ^y¯) for some ¯x, ^y¯ ^∈ ^X. Passing to the limit in condition (5) above and setting P := ∂tEt¯(¯x)+∂tEt¯(¯y) ² > 0, we get that

$$\partialt E{\bar{t}}(\bar{x}) \ge P + \frac{\varepsilon}{2}$$ and $\partialt E{\bar{t}}(\bar{y}) \le P - \frac{\varepsilon}{2}$ . (3.30)

We claim that there exists ^N ^∈ ^N such that for every n > N

$$E_{t_n}(xn) < E{\bar{t}}(\bar{x}) + P(tn - \bar{t})$$ and $E{t_n}(yn) > E{\bar{t}}(\bar{y}) + P(t_n - \bar{t}).$ (3.31)

By symmetry, it is enough to prove the first inequality in (3.31). We then have

$$E_{t_n}(yn) > E{\bar{t}}(\bar{y}) + P(tn - \bar{t}) = E{\bar{t}}(\bar{x}) + P(tn - \bar{t}) > E{t_n}(x_n),$$

which contradicts condition (4) and thus finishes the proof.

To prove Equation (3.31), we choose some ∆x¯, ∆t¯, ε′ , D > 0 such that for all ^t ^∈ [t¯−∆t¯,t¯], x ∈ B∆x¯ (¯x) we have that

$$\partial_t E_t(x) > P + \varepsilon',$$ (3.32)

$$\left| |\partial Et|(x) - |\partial E{\bar{t}}|(x)| \le D|t - \bar{t}|, \tag{3.33} \right|$$

$$E{\bar{t}}(x) - E{\bar{t}}(\bar{x}) \le \varepsilon{\bar{x}} (d(x, \bar{x})) |\partial E{\bar{t}}|(x). \tag{3.34}$$

Here we can demand (3.32) from continuity of ∂tE^t (see Assumption 2) together with Equation (3.30); we can demand (3.33) from the Lipschitz continuity of |∂E^t | in Assump tion 6'; finally, we can demand (3.34) from the condition (2.2) in Assumption 6' applied at the instant ^t¯ and at the point ¯x. For large enough ⁿ, we have that ^tⁿ ^∈ [t¯⁻ ^∆t¯,t¯] and xⁿ ∈ B∆x¯ (¯x). We will now show that ^Et¯(ˆx) + ^P(tⁿ ⁻ ^t¯) ⁻ ^Etⁿ (xn) is positive for large n. Indeed, using conditions (1)–(5) we have that

$$\begin{split} E_{\bar{t}}(\bar{x}) + P(tn - \bar{t}) - E{t_n}(xn) &= \left( E{\bar{t}}(xn) - E{t_n}(x_n) \right) - P(\bar{t} - tn) - \left( E{\bar{t}}(xn) - E{\bar{t}}(\bar{x}) \right) \ &\geq (P + \varepsilon')(\bar{t} - t_n) - P(\bar{t} - tn) - \varepsilon{\bar{x}} \left( d(xn, \bar{x}) \right) |\partial E{\bar{t}}|(x_n) \ &\geq \varepsilon'(\bar{t} - tn) - \varepsilon{\bar{x}} \left( d(x_n, \bar{x}) \right) D(\bar{t} - t_n). \end{split}$$

For large n, εx¯ d(xn, x¯) D becomes arbitrarily small, so that the first term dominates the second. Since ε ′ > 0, this proves inequality (3.31),which results in the contradiction and finishes the proof. □

Remark 16. We observe that, in the proof of Lemma 3.11, we can deduce the existence of N such that ∀n > N : Etⁿ (yn) > Et¯(¯y) + ^P(tⁿ ⁻ ^t¯) by using the condition (2.1) in Assumption 6. However, the latter does not suffice for establishing the relation in Equation (3.31), and we need the strengthen version reported in Assumption 6'.

4. Characterizing energy jumps with actions

As the last step in describing the limit curve ˆη and its energy balance, we would like to characterize the energy jumps Et(ˆη ⁺(t)) ⁻ ^Et(ˆ^η −(t)) and thus ensure that ˆη does not cross energy barriers. The authors of [2] achieved this goal in their setting by describing the energy jumps through a cost function c^t , which is defined by minimizing the energydissipation integrals. In this section, we prove an analogous result for certain transition rules ¯ω by characterizing the energy jumps through actions, which play a similar role as the cost function c in [2, Equation (2.4)], but whose value depends on the specific transition rule ¯ω used in the construction.

4.1. Generalizing from the gradient flow. To investigate transition rules which admit a characterization of the energy jumps, we start by considering transition rules which bring our framework closest to the one in [2], i.e., transition rules arising from the gradient flow, which fulfill the following equation:

$$\bar{\omega}t(x) = \lim{s \to \infty} \gamma(s),$$ (4.1)

where γ is a gradient flow trajectory of E^t starting at x, i.e., γ(0) = x and, for all s > 0,

$$E_t(x) - E_t(\gamma(s)) = \frac{1}{2} \int_0^s |\partial E_t|^2 (\gamma(\tau)) + |\dot{\gamma}(\tau)|^2 d\tau.$$ (4.2)

For such transition rules arising from the gradient flow, we have, for ^x ^∈ ^R n , t ∈ (0, T), t n 1 ↑ t and t n 2 ↓ t,

$$E_t(x) - E_t(\bar{\omega}_t(x)) = c_t^F(x, \bar{\omega}_t(x)) \quad \forall x \in \mathbb{R}^n, \quad \text{and hence (by Equation (3.3))}$$ $$\mu^{\delta}({t}) = ct^F\left(\lim{n \to \infty} \eta^{\delta}(t1^n), \lim{n \to \infty} \eta^{\delta}(t_2^n)\right), \quad (4.3)$$

where c t is defined as follows:

$$c_t^F(u_1, u_2) := \inf \left{ \frac{1}{2} \int_a^b \left( |\partial E_t|^2 (\gamma(s)) + |\dot{\gamma}(s)|^2 \right) \, \mathrm{d}s \, \middle| \, \begin{array}{c} a < b \in \mathbb{R} \ \gamma \in AC([a, b], \mathbb{R}^n) \ \gamma(a) = u_1, \gamma(b) = u_2 \end{array} \right}. \tag{4.4}$$

Let us assume for a moment that we can pass Equation (4.3) to the limit as δ → 0, i.e., that we can prove that, for all δⁿ → 0, t n ¹ → t, t n ² → t,

$$Et\left(\lim{n\to\infty}\eta^{\delta_n}(t_1^n)\right) - Et\left(\lim{n\to\infty}\eta^{\delta_n}(t_2^n)\right) = ct^F\left(\lim{n\to\infty}\eta^{\delta_n}(t1^n), \lim{n\to\infty}\eta^{\delta_n}(t_2^n)\right), \quad \text{and}$$

$$\bar{\mu}({t}) = ct^F\left(\lim{n\to\infty}\eta^{\delta_n}(t1^n), \lim{n\to\infty}\eta^{\delta_n}(t_2^n)\right), \quad (4.5)$$

where we recall that µ ^δⁿ ⇀^∗ µ¯ as in Equation (3.16). In the case (4.5) held, we would immediatly gain two results:

(1) Since c t (x, x′ ) is nonzero whenever x and x ′ belong to different path-connected components of critical points of E^t , Equation (4.5) would immediatly imply that Axiom 3 holds. This would, in turn, guarantee the existence of a limit curve ˆu through Theorem 3.9.
(2) Through the energy balance (3.17) and Equation (4.5), we would obtain the equality

$$\hat{E}\left(\hat{\eta}^{-}(t)\right) - \hat{E}\left(\hat{\eta}^{+}(t)\right) = c_{t}^{F}\left(\eta^{-}(t), \eta^{+}(t)\right),\,$$

where ˆη ±(t) = [(t, η±(t))]. Informally, this implies that the limit curve ˆu does not jump through energy barriers.

Some other interesting transition rules—e.g., those which can be derived from certain discretizations of Equation (4.2)—allow for equalities similar to Equations (4.3) and (4.4) as well, with some modifications. In this section, we develop a framework to deal with those transition rules in a unified way, and to pass the equalities of the form of Equa tion (4.3) to the limit as δ → 0, as in Equation (4.5). On our way to do so, we first describe a sufficient condition to be able to pass to the limit for δ → 0, which has the form of a continuous-in-time triangular inequality of the action c^t in Axiom 3'. Afterwards, we see that this continuous-in-time triangular inequality is fulfilled by actions tailored around Equation (4.4).

Definition 2. An action (ct)t∈[0,T] , henceforth simply written as c^t , is a family of lower semicontinuous functions c^t : ^X ^× ^X ^→ [0, ⁺∞), parametrized by ^t ^∈ [0, T], such that for all x, x′ ^∈ ^X and ^t ^∈ [0, T]:

$$E_t(x) - E_t(x') \le c_t(x, x').$$ (4.6)

A transition rule ¯ω^t is compatible with an action c^t , if for every ^x ^∈ ^X and every ^t ^∈ [0, T]:

$$E_t(x) - E_t(\bar{\omega}_t(x)) = c_t(x, \bar{\omega}_t(x)).$$

Axiom 3'. Let the family of mappings ¯ω^t : ^X ^→ ^X indexed by ^t ^∈ [0, T] be the transition rule as in Definition 1. Then, we require that ¯ω^t is compatible with an action c^t , which has the following property: For all ε, C > ⁰, t ^∈ [0, T], K ^⊆ ^X compact, there exists ∆ > 0 such that:

$$\forall x_1, \dots, x_n \in K, \, \forall t_1, \dots, t_n \in [t - \Delta, t + \Delta] : \ c_t(x_1, xn) \le \sum{i=1}^{n-1} c_{t_i}(xi, x{i+1}) + \varepsilon \quad (4.7)$$

whenever

$$\sum{i=1}^{n-1} c{t_i}(xi, x{i+1}) \le C. \tag{4.8}$$

Furthermore, for x, x′ ^∈ ^C := {^y ^∈ ^X | |∂E^t |(y) = 0}, we have that ct(x, x′ ) = 0 if and only if x and x ′ belong to the same path-connected component of C .

The following lemmas hold.

Lemma 4.1. Axiom 3' implies Axiom 1.

Proof. As the compatibility of ¯ω^t with the action gives Et(x) − E^t ω¯t(x) = c^t x, ω¯t(x) for every ^x ^∈ ^X and for every ^t ^∈ [0, T] (cf. Definition 2), Axiom 1 follows from the nonnegativity of c^t . □

Lemma 4.2. Let us assume Assumptions 1 to 4. Given a non-negative decreasing sequence (δn)ⁿ such that δⁿ → 0 as n → ∞, let η ^δⁿ : [0, T] ^→ ^X be the discrete quasistatic evolutions constructed according to Definition 1 and starting from ^x⁰ ^∈ ^X. Let us further assume that along (η ^δⁿ )ⁿ the convergences reported in (3.16) hold and that Axioms 1, 2 and 3' are satisfied. For every ^t ^∈ [0, T], let us consider ^x1, x² ^∈ ^X and two sequences (t n 1 )n,(t n 2 )ⁿ ⊂ [0, T] such that t n ¹ ≤ t n 2 for every n, t n 1 , tⁿ ² → t as n → ∞, and such that η ^δⁿ (t n 1 ) → x¹ and η ^δⁿ (t n 2 ) → x² as n → ∞. Then,

$$\bar{\mu}({t}) \ge c_t(x_1, x_2), \qquad E_t(x_1) - E_t(x_2) \ge c_t(x_1, x_2),$$ (4.9)

i.e., Axiom 3 holds.

Before we prove Lemma 4.2, let us remark how it characterizes the energy jumps of the limit curve ˆη.

Corollary 4.3. Let us assume Assumptions 1 to 5 and that Axioms 1, 2 and 3' are satisfied. Then, the limit trajectory ηˆ constructed according to Theorem 3.9 fulfills, for all t ∈ [0, T]:

$$\hat{E}(\hat{\eta}^+(t)) - \hat{E}(\hat{\eta}^-(t)) = \bar{\mu}(\lbrace t \rbrace) = \hat{c}_t(\hat{\eta}^+(t), \hat{\eta}^-(t)),$$

where cˆ^t [(t, x)], [(t, x′ )] := ct(x, x′ ).

Proof. The conclusion of Lemma 4.2 is a strengthening of Axiom 3. For the last part of the corollary, note that Axiom 3' implies in particular that c^t fulfills the triangular inequality for a fixed t. Together with the last part of Axiom 3', this ensures that ˆc^t

factors through X as described by ˆc^t . Combining Equation (4.6) with Theorem 3.9, we get that

Eˆ ηˆ ⁺(t) ⁻ ^E^ˆ ηˆ −(t) = ¯µ({t}) ≤ cˆ^t ηˆ ⁺(t), ηˆ −(t) .

Further combining this inequality with the reverse inequality in Equation (4.9) finishes the proof. □

Proof of Lemma 4.2. Let us fix t ∈ [0, T] and ε > 0. From Lemma 3.3 we have that there exists a compact set ^K ^⊂ ^X such that Im(^η ^δⁿ ) ^⊂ ^K for every ⁿ ^∈ ^N. Furthermore, by Lemma 3.1 and since E and ∂tE are bounded in K, we can choose C such that µ ^δⁿ ((0, T]) ^≤ ^C for all ⁿ. In particular, we notice that for every ⁿ

$$\sum{t'{i} \in J^{\delta{n}}} c{t'{i}} \left( \eta^{\delta{n}}(t'{i}), \eta^{\delta{n}}(t'{i+1}) \right) \leq \mu^{\delta{n}}((0, T]) \leq C.$$

Let ∆ > 0 be as in Axiom 3'.For any t1, t² such that t − ∆ ≤ t¹ < t² ≤ t + ∆ we thus have that

$$c_t\left(\eta^{\delta_n}(t_1), \eta^{\delta_n}(t2)\right) \leq \sum{i=1}^{n-1} c_{t_i'}\left(\eta^{\delta_n}(t_i'), \eta^{\deltan}(t{i+1}')\right) + \varepsilon$$ $$= \mu^{\delta_n}((t_1, t_2]) + \varepsilon,$$

where t ⁱ ∈ J ^δⁿ are the jump points of η ^δⁿ in the interval (t1, t2]. Hence, for all 0 ^< ^∆′ ^≤ ^∆ it holds

$$\begin{split} \bar{\mu}([t-\Delta',t+\Delta']) &\geq \limsup_{n\to\infty} \mu^{\deltan}([t-\Delta',t+\Delta']) \geq \limsup{n\to\infty} \mu^{\delta_n}([t_1^n,t2^n]) \ &\geq \limsup{n\to\infty} c_t \left(\eta^{\delta_n}(t_1^n),\eta^{\delta_n}(t_2^n)\right) - \varepsilon \ &\geq c_t(x_1,x_2) - \varepsilon, \end{split}$$

where we used the lower semicontinuity of c^t in the last line. If we let ∆′ → 0, we see that ¯µ({t}) ≥ ct(x1, x2) − ε. Taking the limit as ε → 0, we get the first inequality in (4.9). For the second one, we observe that

$$\begin{split} E_t(x_1) - E_t(x2) &= \lim{n \to \infty} \left( E_{t_1^n} \left( \eta^{\delta_n}(t1^n) \right) - E{t_2^n} \left( \eta^{\delta_n}(t2^n) \right) \right) = \lim{n \to \infty} \left( \mathcal{E}_{t_1^n} (\eta^{\deltan}) - \mathcal{E}{t_2^n} (\eta^{\deltan}) \right) \ &= \lim{n \to \infty} \left( \int_{t_1^n}^{t_2^n} \mathcal{D}^{\delta_n}(s) \, \mathrm{d}s + \mu^{\delta_n} ((t_1^n, t2^n]) \right) = \lim{n \to \infty} \mu^{\delta_n} ((t_1^n, t_2^n]), \end{split}$$

and we conclude by using the same arguments as before. □

To construct actions fulfilling Axiom 3', we will now investigate a possible proof strategy to show that the transition rule arising from the gradient flow, as in Equations (4.1) to (4.4), fulfills Axiom 3'. Discussing the proof strategy will allow us to distill sufficient properties for more general transition rules for which Axiom 3' holds true.

The resemblance of Equation (4.7) in Axiom 3' to the triangular inequality already suggests a strategy to show that the transition rule (4.1) corresponding to the gradient flow fulfills Axiom 3': Similar to the usual approach for proving the triangular inequality, we could attempt to concatenate near-optimal curves from xⁱ to xi+1 to construct a competitor curve from x¹ to xn. At first glance, this na¨ıve proof strategy appears promising. Suppose that near-optimal curves γⁱ : [aⁱ , bⁱ ] ^→ ^R ⁿ are chosen such that bⁱ = ai+1 and denote their concatenation by ¯γ. From Equation (4.4), we have

$$\begin{split} c_t(x_1,xn) &\leq \frac{1}{2} \int{a1}^{b{n-1}} \left( |\partial Et|^2 (\bar{\gamma}(s)) + |\dot{\bar{\gamma}}(s)|^2 \right) \, \mathrm{d}s \ &= \sum{i=1}^{n-1} \frac{1}{2} \int_{a_i}^{b_i} \left( |\partial E_t|^2 (\gamma_i(s)) + |\dot{\gamma}i(s)|^2 \right) \, \mathrm{d}s \ &= \sum{i=1}^{n-1} \frac{1}{2} \int_{a_i}^{bi} \left( |\partial E{t_i}|^2 (\gamma_i(s)) + |\dot{\gamma}i(s)|^2 \right) \, \mathrm{d}s \ &+ \underbrace{\sum{i=1}^{n-1} \frac{1}{2} \int_{a_i}^{b_i} (|\partial E_t|^2 (\gammai(s)) - |\partial E{t_i}|^2 (\gammai(s))) \, \mathrm{d}s}{\varepsilon'} \ &\leq \sum{i=1}^{n-1} \left( c{t_i}(xi, x{i+1}) + \varepsilon_i \right) + \varepsilon'. \end{split}$$

Here, εⁱ can be controlled by selecting near-optimal curves γⁱ , and ε ′ can be managed by choosing ∆ sufficiently small. However, it is impossible to select a ∆ that makes ε ′ uniformly small because we cannot control the time spans bⁱ − aⁱ of the curves γⁱ . Moreover, ε ′ is expected to grow with n, yet we require a ∆ that is effective for all n. To be able to control ε ′ , we need to modify ¯γ such that it does not spend too much time in areas where |∂E^t | 2 is significantly larger than |∂Etⁱ | 2 . This way, we hope to ensure that ε ′ is bounded by a constant.

To modify ¯^γ to enable such a bound, our initial idea is to partition ^K′ ^⊆ ^X—where ^K′ is some compact such that Im(¯γ) ⊆ K′—into two distinct regions:

Expensive Region: Areas where ¹ 2 |∂E^t |(x)—and, for sufficiently small ∆, also 2 |∂Etⁱ | 2 (x)—exceeds a certain threshold Pexp.

Cheap Region: Areas where ¹ 2 |∂E^t | 2 (x) is below a threshold Pcheap.

We leave ¯γ unchanged in the expensive region, following along the curves γⁱ . Whenever γ¯ enters the cheap region, we take a shortcut to the point where ¯γ exits the cheap region (see Figure 3). The total time spent in the expensive region by ¯γ is bounded, which consequently bounds ε ′ . This is because, if we call S i exp ⊆ [aⁱ , bⁱ ] the time spent in the expensive region by γⁱ , we have, using the bound C assumed in Axiom 3' in Equation (4.8):

$$\left| \bigcup{i} S{\text{exp}}^{i} \right| = \frac{1}{P{\text{exp}}} \sum{i} \int{S{\text{exp}}^{i}} P{\text{exp}} \, \mathrm{d}s \le \frac{1}{P{\text{exp}}} \sum{i} \int{S{\text{exp}}^{i}} \frac{1}{2} |\partial E{t{i}}|^{2} (\gamma{i}(s)) \, \mathrm{d}s$$

$$\le \frac{1}{P{\text{exp}}} \sum{i} \frac{1}{2} \int{a{i}}^{b{i}} \left( |\partial E{t{i}}|^{2} (\gamma{i}(s)) + |\dot{\gamma}_{i}(s)|^{2} \right) \, \mathrm{d}s$$

$$\le \frac{1}{P{\text{exp}}} \sum{i} \left( c{t{i}}(x{i}, x{i+1}) + \varepsilon{i} \right) \le \frac{C + \sum{i} \varepsilon{i}}{P{\text{exp}}}.$$

Since we can control εⁱ by choosing near-optimal curves γⁱ , we can find a bound of the total time spent in the expensive region—independendly of n. Thus, we can make ε ′ arbitrarily small by choosing ∆ sufficiently small. However, controlling the cost of shortcuts through the cheap region remains challenging.

To address this, we introduce a strategy of how to choose the two regions: We enlarge the set of critical points in K—where |∂E^t | = 0—by a radius r, forming a cheap region C ^r where ¹ 2 |∂E^t | 2 (x) ≤ Pcheap := maxx∈^C ^r 1 2 |∂E^t | 2 (x). Traversing through sets of critical points incurs no cost, which means that the cost of each shortcut is determined by the part of the shortcut which does not go through of critical points. By taking the shortest route to and from the next critical point at the beginning and the end of each shortcut, respectively, we can control the cost of each shortcut by r and Pcheap. Furthermore, we can make both Pcheap and r small by choosing r: Due to the continuity of |∂E^t |, Pcheap → 0 as r → 0.

What remains to be bounded is the number of shortcuts. Because the set of critical points can be expressed as the disjoint union of well-seperated compacts by Assumption 5, K contains a finite number of connected components of critical points. Enlarging each of these components by r does not increase the number of connected components, so C ^r contains a finite number of connected components as well. Additionally, by consistently taking a shortcut to the last point where ¯γ exits a connected component of C ^r , we ensure that each connected component of C ^r is traversed at most once — limiting the number of shortcuts.

Finally, leveraging the continuity of |∂E^t |, 1 2 |∂E^t | 2 (x) is bounded below in K′ \C ^r , which makes K′ \ C ^r an expensive region and enables the application of our initial strategy.

Figure 3: The proof strategy for Lemma 4.8 and the scheme for shortening competitor curves: We create a shortcut from the first point where a curve enters a neighborhood Br(Ci) to the last point where one exits Br(Ci), where Cⁱ is a connected component of critial points. This approach ensures that Br(Ci) is traversed at most once. In the formal proof, rather than constructing a new competitor curve by concatination, we employ the triangular inequality to estimate ^ct(X1, X4) ^≤ ^ct(X1, p¹ in) + ct(p 1 in, p 1 in) + ct(p 1 in, p out) + ct(p 1 out, p¹ out) + ct(p out, X4).

From this discussion, we are now in a position to distill the essential properties of the gradient flow that allow for a proof of Axiom 3'. As those properties are satisfied by a range of transition rules —whether they are discrete or additionally incorporate momentum—, the application of the resulting framework is not limited to the gradient flow. To allow for a unified exposition of such actions, we consider curves in the phase space. As a guide to the formal proof, Figure 3 should be understood as a schematic representation of this phase space.

In the following, we denote (a, b] ^∩ ^I and [a, b] ^∩ ^I as (a, b]^I and [a, b]^I , respectively, whenever ^I ∈ {R, ^Z} and a, b ^∈ ^I.

Definition 3. An action c^t is generated by curves if there exists an index set ^I ∈ {Z, ^R}, a phase space P equipped with a metric d^P and functions

$$\begin{array}{cccccccccc$$

such that for every γ ∈ Γ(p1, p2) it holds γ(a) = p¹ and γ(b) = p2, i is a closed immersion, L is continuous, and c^t : ^X ^× ^X ^→ [0, ^∞) is given as

$$\begin{split} c_t(x_1, x_2) &= \mathfrak{c}_t(i(x_1), i(x_2)) &\quad \text{for } x_1, x_2 \in \mathbb{X}, \ \mathfrak{c}_t(p_1, p2) &:= \inf{\gamma \in \Gamma(p_1, p2)} \int{(a, b]_{\mathbb{I}}} \mathcal{L}_t(\gamma_s) \, \mathrm{d}\nu(s) &\quad \text{for } p_1, p_2 \in \mathscr{P}, \end{split} \tag{4.10}$$

where ν is the Lebesgue measure if I = R and the counting measure if I = Z. We call the value of Γ the set of admissible curves for p1, p² ∈ P, and the value of L the price for p ∈ P and t ∈ [0, T].

Moreover, we require that the following properties hold:

Property 1: Restrictions, concatenations and shifts of admissible curves are again admissible curves. In other words, for ^p1, p2, p³ ^∈ ^P and ^a ^≤ ^s ^≤ ^b ^∈ ^I, the following holds:

Restrictions: For all γ ∈ Γ(p1, p3) such that Dom(γ) = [a, b]^I and γ(s) = p2, we have that γ|[a,s]^I ∈ Γ(p1, p2) and γ|[s,b]^I ∈ Γ(p2, p3).
Concatenations: For all γ ∈ Γ(p1, p2) and γ ′ ∈ Γ(p2, p3) such that Dom(γ) = [a, s]I and Dom(γ ′ ) = [s, b]I , we have that γ ∪ γ ′ ∈ Γ(p1, p3), where γ ∪ γ ′ [a, b]^I → P is defined as

$$\gamma \cup \gamma'(s) \coloneqq \begin{cases} \gamma(s) & \text{if } s \in [a,s]{\mathbb{I}}, \ \gamma'(s) & \text{if } s \in (s,b]{\mathbb{I}}. \end{cases}$$

• Shifts: For all ^γ ^∈ Γ(p1, p2) such that Dom(γ) = [a, b] and all ^r ^∈ ^I, we have that γ ′ : [a − r, b − r] → P with γ ′ (t) = γ(t + r) is in Γ(p1, p2).

Property 2: For ^p ^∈ ^P, ^Lt(p) = 0 if and only if there exists ^x ^∈ ^X such that ⁱ(x) = ^p and |∂E^t |(x) = 0.

Property 3: For ^x1, x² ^∈ ^X with ^x¹ ̸⁼ ^x2, ^ct(x1, x2) = 0 if and only if ⁱ(x1) and ⁱ(x2) lie in the same connected component of {p ∈ P | Lt(p) = 0}.

Property 4: For all K ⊆ P compact, there exists some K′ ⊆ P compact such that for all p1, p² ∈ K and all ε > 0, there exists γ : [a, b]^I → P, γ ∈ Γ(p1, p2) such that Im(γ) ⊆ K′ and

$$\int{(a,b]{\mathbb{I}}} \mathcal{L}_t(\gamma(s)) d\nu(s) \le \mathfrak{c}_t(p_1,p_2) + \varepsilon.$$

This means that for points p1, p² ∈ K, we may take the infimum in Equation (4.10) over only those admissible curves whose image lies in K′ .

Property 5: For all K ⊆ P compact and ε > 0, there exists some L, P > 0 such that for all p1, p² ∈ K such that Lt(p1) ≤ P, Lt(p2) ≤ P, and dP(p1, p2) ≤ L, we have that ct(p1, p2) ≤ ε. Informally, this means that the value of the action is small between two points which are close to each other and where L^t is small.

Remark 17. Notice that in what follows, the expressions of the functions appearing in Definition 3 will be energy-dependent.

4.2. Examples of transition rules. Before we show how the abstract Definition 3 relates to Axiom 3', we present some examples of actions—and the corresponding transition rules—that are generated by curves. The fact that those actions indeed satisfy the required properties is contained in Propositions 4.4 to 4.6, whose proofs are postponed to Section 5.

4.2.1. The gradient flow.

Definition 4. A transition rule of the gradient flow is a transition rule ¯ω F t : ^X ^→ ^X for which there exists, for each ^x ^∈ ^X, a curve ^ϕ ^∈ AC([0, ^∞), ^X) and a sequence (sk)k≥¹ with s^k ↗ +∞ as k → ∞ such that ϕ(0) = x, limk→∞ d ϕ(sk), ω¯ t (x) = 0, and, for all s ∈ [0, ∞),

$$E_t(x) - E_t(\phi(s)) = \frac{1}{2} \int_0^s |\partial E_t|^2 (\phi(\sigma)) + |\dot{\phi}(\sigma)|^2 d\sigma.$$

The action of the gradient flow is the action c ^t given by

$$c_t^F(x_1, x_2) := \inf \left{ \frac{1}{2} \int_a^b \left( |\partial E_t|^2 (\phi(s)) + |\dot{\phi}(s)|^2 \right) ds \, \middle| \, \begin{array}{c} a < b \in \mathbb{R} \ \phi \in AC([a, b], \mathbb{X}) \ \phi(a) = x_1, \phi(b) = x_2 \end{array} \right}. \tag{4.11}$$

Proposition 4.4. Let us assume Assumptions 1, 2, 4, 5 and 7. A transition rule ω¯ F t of the gradient flow is compatible—as defined in Definition 2—with the action c F ^t of the gradient flow. Furthermore, the action c F t is generated by curves as per Definition 3, where I = R and where we use the following definitions:

$$\mathscr{P}^F := \mathbb{X} \times \mathbb{R}^+, \text{ where } d_{\mathscr{P}^F} \left( (x_1, v_1), (x_2, v_2) \right) = d(x_1, x_2) + |v_1 - v_2| \tag{4.12}$$

$$i^F(x) \coloneqq (x,0) \tag{4.13}$$

$$\Gamma(p_1, p_2) := \left{ \gamma \colon [a, b] \to \mathscr{P}^F \middle| \begin{array}{l} a < b \in \mathbb{R}, \gamma(a) = p_1, \gamma(b) = p_2, \ \gamma_x \in AC([a, b]; \mathbb{X}), |\dot{\gamma}_x| \equiv \gamma_v \, a.e. \end{array} \right}$$

$$(4.14)$$

$$\mathcal{L}{t}^{F}\left((x,v)\right) \coloneqq \frac{1}{2}|\partial E{t}|^{2}(x) + \frac{1}{2}v^{2} \tag{4.15}$$

$$\mathbf{c}_t^F(p_1, p2) \coloneqq \inf{\gamma \in \Gamma(p_1, p_2)} \int_a^b \mathcal{L}_t^F(\gamma(s)) \, \mathrm{d}s. \tag{4.16}$$

Here, for every interval ^I ^⊆ ^R, we denote the components of a curve ^γ : ^I ^→ ^P^F by ^γ^x : ^I ^→ ^X and ^γ^v : ^I ^→ ^R +.

4.2.2. The minimizing movement scheme.

Definition 5. For τ > 0, ^x ^∈ ^X and ^t ^∈ [0, T], a minimizing movement scheme sequence starting at point ^x at time ^t is a sequence (us)s∈^N in ^X such that ^u⁰ ⁼ ^x, and, for all ^s ^∈ ^N,

$$u{s+1} \in \mathop{\arg\min}{y \in \mathbb{X}} \left{ E_t(y) + \frac{1}{2\tau} d(y, u_s)^2 \right}.$$

A transition rule of the minimizing movement scheme is a transition rule ^τ ω¯ M t : ^X ^→ ^X for which there exists, for each ^x ^∈ ^X and ^t ^∈ [0, T], a minimizing movement scheme sequence (us)s∈^N in ^X starting at point ^x at time ^t such that ^τ ω¯ M t (x) is a cluster point of (us)s∈N. The action of the minimizing movement scheme is the function ^τ c t : ^X ^× ^X ^→ [0, ^∞) given by

$$\overline{c}{t}^{M}(x{1}, x{2}) := \inf \left{ \sum{s=a}^{b} E{t}(u{s}) - E{t, \tau}^{M}(u{s}) + \frac{1}{2\tau} d(u{s+1}, u{s})^{2} \left| \begin{array}{l} a \le b \in \mathbb{Z}, \ (u{s}){s \in [a, b]{\mathbb{Z}}}, \ u{a} = x{1}, u{b} = x_{2} \end{array} \right},$$ (4.17)

where we set ub+1 = u^b = x² and

$$E{t,\tau}^{M}(x) := \inf{y \in \mathbb{X}} \left{ E_{t}(y) + \frac{1}{2\tau} d(x,y)^{2} \right}. \tag{4.18}$$

We mention that in [25, Definition 5.14] the authors introduced an action for the minimizing movement scheme in Euclidean setting which is fully compatible with Equa tions (4.17) and (4.18).

Proposition 4.5. Let us assume Assumptions 1 to 5 and 7. For τ > 0, a transition rule τ ω¯ M ^t of the minimizing movement scheme is compatible—as defined in Definition 2—with the action ^τ c M ^t of the minimizing movement scheme.

Moreover, if τ ≤ 1 L , the map ^τ c t : ^X ^× ^X ^→ [0, ^∞) is an action generated by curves as per Definition 3, where I = Z and where we use the following definitions:

$$\mathscr{P}^{M} := \mathbb{X} \times \mathbb{X}, \text{ where } d{\mathscr{P}^{M}}(\underline{x},\underline{x}') = \frac{1}{2} \left( d(x{0},x'{0}) + d(x{1},x'_{1}) \right) \tag{4.19}$$

$$i^{M}(x) \coloneqq (x, x) \tag{4.20}$$

$$\Gamma(\underline{x},\underline{x}') := \left{ \gamma \colon [a,b]{\mathbb{Z}} \to \mathscr{P}^{M} \middle| \begin{array}{l} a < b \in \mathbb{Z}, \gamma(a) = \underline{x}, \gamma(b) = \underline{x}', \ \gamma{1}(s) = \gamma{0}(s+1) \text{ for } s \in [a,b-1]{\mathbb{Z}} \end{array} \right}$$

$$(4.21)$$

$$\mathcal{L}{t}^{M}(\underline{x}) := E{t}(x{0}) - E{t,\tau}^{M}(x{0}) + \frac{1}{2\tau}d(x{0}, x_{1})^{2}, \tag{4.22}$$

$${}^{\tau}\mathfrak{c}t^{\scriptscriptstyle{M}}(\underline{x},\underline{x}') \coloneqq \inf{\gamma \in \Gamma(\underline{x},\underline{x}')} \sum_{s=a+1}^b \mathcal{I}_t^{\scriptscriptstyle{M}}(\gamma(s)), \tag{4.23}$$

where E t,τ is defined as in Equation (4.18). Here, for every ^I ^⊆ ^Z we denote the components of a curve γ : I → P^M by γ⁰ and γ1, and we denote the components of points x, x′ ^∈ ^P^M ⁼ ^X ^× ^X by (x0, x1) := ^x and (^x 0 , x′ 1 ) := x ′ .

4.2.3. The BDF2 method for the gradient flow.

Definition 6. For τ > 0, ^x ^∈ ^X and ^t ^∈ [0, T], a BDF2 discretization sequence starting at point ^x at time ^t is a sequence (us)s∈^N in ^X such that ^u⁰ ⁼ ^x and, for all ^s ^∈ ^N,

$$u_{s+1} \in \underset{y \in \mathbb{X}}{\operatorname{arg \, min}} \left{ E_t(y) + \frac{1}{\tau} d(y, us)^2 - \frac{1}{4\tau} d(y, u{s-1})^2 \right},$$

where we set u−¹ = x. A transition rule of the BDF2 discretization is a transition rule τ ω¯ t : ^X ^→ ^X for which there exists, for each ^x ^∈ ^X and ^t ^∈ [0, T], a BDF2 discretization sequence (us)s∈^N in ^X starting at point ^x at time ^t such that ^τ ω¯ B t (x) is a cluster point of (us)s∈N. The action of the BDF2 discretization is the map ^τ c B t : ^X ^× ^X ^→ [0, ^∞) given by

$$\tau{c{t}}^{B}(x{1}, x{2}) := \inf \left{ \sum{s=a}^{b+1} \left( E{t}(u{s}) - E{t, \tau}^{B}(u{s}, u{s-1}) + \frac{1}{2\tau} d(u{s+1}, u{s})^{2} + \frac{1}{2\tau} d(u{s}, u{s-1})^{2} - \frac{1}{4\tau} d(u{s+1}, u{s-1})^{2} \right) \left| \begin{array}{c} a \le b \in \mathbb{Z}, \ (u{s}){s \in [a, b]{\mathbb{Z}}}, \ u{a} = x{1}, u{b} = x_{2} \end{array} \right},$$

$$(4.24)$$

where where we set ua−¹ = x1, ub+1 = ub+2 = x² and

$$E{t,\tau}^{B}(x,x') := \inf{y \in \mathbb{X}} \left{ E_{t}(y) + \frac{1}{\tau} d(y,x)^{2} - \frac{1}{4\tau} d(y,x')^{2} \right}. \tag{4.25}$$

To the best of our knowledge, the action for the BDF2 scheme provided by Equa tions (4.24) and (4.25) is completely original.

Proposition 4.6. Let us sassume Assumptions 1 to 5 and 7. For τ > 0, a transition rule ^τ ω¯ B ^t of the BDF2 discretization is compatible—as defined in Definition 2—with the action ^τ c B ^t of the BDF2 discretization. Moreover, for every τ ≤ 1 L , the map ^τ c B t : ^X×^X ^→ [0, ^∞) is an action generated by curves

as per Definition 3, where I = Z and where we use the following definitions:

$$\mathscr{P}^{B} := \mathbb{X} \times \mathbb{X} \times \mathbb{X}, \text{ where } d{\mathscr{P}^{B}}(\underline{x}, \underline{x}') = \frac{1}{3} \sum{i=1}^{1} d(x{i}, x'{i})$$

$$(4.26)$$

$$i^B(x) := (x, x, x)$$ (4.27)

$$\Gamma(\underline{x},\underline{x}') := \left{ \gamma \colon [a,b]{\mathbb{Z}} \to \mathscr{P}^{B} \middle| \begin{array}{c} a < b \in \mathbb{Z}, \gamma(a) = \underline{x}, \gamma(b) = \underline{x}', \ \gamma{1}(s-1) = \gamma{0}(s) = \gamma{-1}(s+1) \text{ for } s \in [a+1,b-1]_{\mathbb{Z}} \end{array} \right}$$

$$(4.28)$$

$$\mathcal{L}{t}^{B}(\underline{x}) := E{t}(x{0}) - E{t,\tau}^{B}(x{0}, x{-1}) + \frac{1}{2\tau} \left( d(x{0}, x{1})^{2} + d(x{-1}, x{0})^{2} \right) - \frac{1}{4\tau} d(x{-1}, x{1})^{2}, \tag{4.29}$$

$$\mathcal{T}{t}^{B}(\underline{x},\underline{x}') := \inf{\gamma \in \Gamma(\underline{x},\underline{x}')} \sum{s=a+1}^{b} \mathcal{T}{t}^{B}(\gamma(s)), \tag{4.30}$$

where E t,τ is defined as in Equation (4.25). Here, for every ^I ^⊆ ^Z, we denote the components of a curve γ : I → P^B by γ−1, γ⁰ and γ1, and we denote the components of points x, x′ ^∈ ^P^B ⁼ ^X ^× ^X ^× ^X by (x−1, x0, x1) := ^x and (^x ′ −1 , x′ 0 , x′ 1 ) := x ′ .

4.3. A rigorous proof of Axiom 3'. With those examples in mind, we now show that actions generated by curves fulfill Axiom 3'. We postpone the proof of the following ancillary lemma to the end of this section.

Lemma 4.7. Let c^t : ^X×^X ^→ [0, ^∞) be an action generated by curves as per Definition 3 and let c^t : P × P → [0, ∞) be the map defined in (4.10). Then, c^t and c^t fulfill the triangular inequality, i.e., for every ^p1, p2, p³ ^∈ ^P, every ^x1, x2, x³ ^∈ ^X, and every t ∈ [0, T] we have

$$\mathfrak{c}_t(p_1, p_3) \le \mathfrak{c}_t(p_1, p_2) + \mathfrak{c}_t(p_2, p_3)$$ and $c_t(x_1, x_3) \le c_t(x_1, x_2) + c_t(x_2, x_3)$ .

We are now ready to prove the key lemma of this section.

Lemma 4.8. Let us assume Assumptions 1 to 5. If a transition rule ω¯^t is compatible with an action c^t generated by curves, then Axiom 3' holds.

Proof. We follow the idea outlined on pages 26–28 and illustrated in Figure 3. In the first part of the proof, we obtain a ∆ > 0 as demanded by Axiom 3'. In the second part, we confirm that with such a choice of ∆ > 0 and for any x1, . . . , xn, t1, . . . , tⁿ as in Axiom 3', Equation (4.7) is indeed fulfilled.

Let ε, C > 0, ^t ^∈ [0, T] and ^K ^⊆ ^X compact be given as in Axiom 3'. We first notice that i(K) is compact, as i is continuous. Hence, we may find a compact set K′ ⊆ P related to i(K) as prescribed by Property 4. By Property 2 and Assumption 5, since i is a closed immersion, the connected components of {p ∈ K′ | Lt(p) = 0} are well-separated, and hence there are only finitely many of them. We enumerate those connected components as C1, . . . , C^m and use Property 5 to choose L, Pcheap > 0 such that for all p1, p² ∈ K′ with Lt(p1) ≤ Pcheap, Lt(p2) ≤ Pcheap and dP(p1, p2) ≤ L, we have that

$$c_t(p_1, p_2) \le \frac{\varepsilon}{6(m+1)}. (4.31)$$

For r > 0, we write C ^r := S^m ^j=1 Br(C^j ), where Br(C^j ) := {p ∈ P | dP(p, C^j ) < r}. By the continuity of L and the compactness of K′ , supp∈(^C ^r Lt(p) → 0 as r → 0. Thus, we can choose 0 < r < L such that supp∈(^C ^r Lt(p) ≤ Pcheap. Moreover, by further reducing r if needed, we can assume that Br(C^j ) ∩ Br(C^j ′) = ∅ whenever j ̸= j ′ . Using a similar argument, we see that

$$\inf_{p \in K' \setminus \underline{\mathscr{C}}_r} \mathcal{L}_t(p) > 0.$$

Even more, we can choose some ∆′′ such that

$$\Delta{\mathcal{L}} := \inf{\substack{p \in K' \setminus \mathscr{C}r \ t' \in [t - \Delta'', t + \Delta'']}} \mathcal{L}{t'}(p) > 0.$$

$$(4.32)$$

Using continuity one last time, we can choose some ∆′ > 0 such that for all p ∈ K′ and t ′ ∈ [t − ∆′ , t + ∆′ ], we have that

$$|\mathcal{L}{t'}(p) - \mathcal{L}{t}(p)| \le \varepsilon{\mathcal{L}} := \frac{\varepsilon}{3} \cdot \left(\frac{C + \frac{\varepsilon}{3}}{\Delta{\mathcal{L}}} + 2(m+1)\right)^{-1}.$$ (4.33)

We claim that ∆ := min{∆′ , ∆′′} is the ∆ > 0 desired by Axiom 3'.

To prove this, we fix points x1, . . . , xⁿ ∈ K and instants t1, . . . , tⁿ ∈ [t − ∆, t + ∆]

such that Pn−¹ ⁱ=1 ctⁱ (xⁱ , xi+1) ≤ C. We first choose γⁱ ∈ Γ i(xi), i(xi+1) according to Property 4 such that Im(γi) ⊆ K′ and

$$\int_{(a_i,bi]{\mathbb{I}}} \mathcal{L}_{t_i}(\gammai(s)) \,\mathrm{d}\nu(s) \le c{t_i}(xi,x{i+1}) + \frac{\varepsilon}{3n},\tag{4.34}$$

where ^aⁱ < bⁱ ^∈ ^I are such that Dom(γi) = [aⁱ , bⁱ ]I . We set IS := {(i, s) | i ∈ {1, . . . , n}, s ∈ [aⁱ , bⁱ ]I} and

$$IS_{\text{cheap}} := \left{ (i, s) \in IS \mid \gamma_i(s) \in \overline{(\underline{\mathscr{C}}_r)} \right}.$$

Note that for all ⁱ, {^s ^∈ ^I ^| (i, s) ^∈ IScheap} is closed and that for all (i, s) ^∈ IS\ISc, we have by Equation (4.32) that

$$\mathcal{L}_{t_i}(\gammai(s)) \ge \Delta{\mathcal{L}}.\tag{4.35}$$

For the ease of notation, we use the lexicographic order on the pairs (i, s) ∈ IS, where (i, s) < (j, s′ ) if i < j or if i = j and s < s′ . For (i, s) ∈ IScheap, we also set J(i, s) to be the unique index j ∈ {1, . . . , m} such that γi(s) ∈ Br(C^j ).

As a next step, we inductively define breakpoints, which we will eventually use to get an upper bound on ct(x1, x2) by repeatedly using the triangular inequality. We start by setting

$$(i{\text{out}}^{0}, s{\text{out}}^{0}) := \begin{cases} \max{(i, s) \in IS{\text{cheap}} \mid J(i, s) = J(1, a{1})} & \text{if } (1, a{1}) \in IS{\text{cheap}}, \ (1, a_{1}) & \text{otherwise,} \end{cases}$$ (4.36)

and inductively set, for each k

$$\begin{split} &(i{\text{in}}^{k+1}, s{\text{in}}^{k+1}) \coloneqq \min{(i, s) \in IS{\text{cheap}} \mid (i, s) > (i{\text{out}}^k, s{\text{out}}^k)}, \ &(i{\text{out}}^{k+1}, s{\text{out}}^{k+1}) \coloneqq \max{(i, s) \in IS{\text{cheap}} \mid J(i, s) = J(i{\text{in}}^k, s{\text{in}}^k)}. \end{split}$$

We terminate this scheme at step d if there is no possible choice for (i d+1 in , sd+1 in ). Formally, we set (i d+1 in , sd+1 in ) = (n, bn) and (i 0 in, s⁰ in) = (1, a1). The resulting breakpoints have the following properties:

(1) d ≤ m (where m is the number of components C1, . . . , Cm).
(2) (i k out, s^k out) ∈ IScheap and (i k in, s^k in) ∈ IScheap for all 1 ≤ k ≤ d.
(3) J(i k in, s^k in) = J(i k out, s^k out) for all 1 ≤ k ≤ d.
(4) For all (i k out, s^k out) < (i, s) < (i k+1 in , sk+1 in ), we have that (i, s) ∈/ IScheap.

We set, for 0 ≤ k ≤ d+ 1, p k in := γⁱ k in (s k in) and p k out := γⁱ k out (s k out). In the rest of this proof, we first estimate ct(p k out, pk+1 in ), then ct(p k in, p^k out), and then we combine those estimates using the triangular inequality to get an upper bound on ct(x1, xn).

Estimating ct(p k out, pk+1 in ): We set, for 1 ≤ i ≤ n − 1 and 0 ≤ k ≤ d,

$$\begin{split} S_{\text{exp}}^{i,k} &:= {s \in (a_i,bi]{\mathbb{I}} \mid (i{\text{out}}^k, s{\text{out}}^k) < (i,s) < (i{\text{in}}^{k+1}, s{\text{in}}^{k+1}) } \ S_{\text{exp}}^{i,k} &:= {s \in (a_i,bi]{\mathbb{I}} \mid (i{\text{out}}^k, s{\text{out}}^k) \leq (i,s) \leq (i{\text{in}}^{k+1}, s{\text{in}}^{k+1}) }, \end{split}$$

where some of the S i,k exp and S¯i,k exp might be empty. By item 4 above and Equation (4.35), we see that for all 1 ≤ i ≤ n − 1, 0 ≤ k ≤ d and s ∈ S i,k exp, we have that Ltⁱ (γi(s)) ≥ ∆L. Thus, we have

$$C + \frac{\varepsilon}{3} \ge \sum{i=0}^{n-1} \int{(a{i},b{i}]{\mathbb{I}}} \mathcal{L}{t{i}}(\gamma{i}(s)) \, d\nu(s) \ge \sum{k=0}^{d} \sum{i=i{\text{out}}^{k}}^{\sum{i=1}^{k}} \int{S{\text{exp}}^{i,k}} \mathcal{L}{t{i}}(\gamma_{i}(s)) \, d\nu(s)$$

$$\ge \sum{k=0}^{d} \sum{i=i{\text{out}}^{k+1}}^{i{\text{in}}^{k+1}} \int{S{\text{exp}}^{i,k}} \Delta{\mathcal{L}} \, d\nu(s) = \Delta{\mathcal{L}} \cdot \sum{k=0}^{d} \sum{i=i{\text{out}}^{k}}^{i{\text{in}}^{k+1}} \nu(S_{\text{exp}}^{i,k}).$$

From the chain of inequalities above and the fact that for a fixed 0 ≤ k ≤ d, we have that Pⁱ k+1 in i=i k out ν(S i,k exp) − ν(S i,k exp) ≤ 2,we get

$$\varepsilon{\mathcal{L}} \cdot \left( \sum{k=0}^{d} \sum{i=i{\text{out}}^{k+1}}^{i{\text{in}}^{k+1}} \nu(\overline{S}{\text{exp}}^{i,k}) \right) \leq \varepsilon{\mathcal{L}} \cdot \left( \sum{k=0}^{d} \left( \sum{i=i{\text{out}}^{k}}^{i{\text{in}}^{k+1}} \nu(S{\text{exp}}^{i,k}) + 2 \right) \right) \ \leq \varepsilon{\mathcal{L}} \cdot \left( \frac{C + \frac{\varepsilon}{3}}{\Delta{\mathcal{L}}} + 2(m+1) \right) = \frac{\varepsilon}{3}. \tag{4.37}$$

With this estimate, we have that

$$\begin{split} & \sum{k=0}^{d} \mathfrak{c}{t}(p{\text{out}}^{k}, p{\text{in}}^{k+1}) \leq \sum{k=0}^{d} \left( \sum{i=i{\text{out}}^{k}}^{\sum{i=i{\text{out}}}^{k+1}} \int{\overline{S}{\text{exp}}^{i,k}} \mathcal{L}{t}(\gamma{i}(s)) \, \mathrm{d}\nu(s) \right) \ & \leq \sum{i=1}^{n-1} \underbrace{\int{(a{i}, b{i}]{\mathbb{I}}} \mathcal{L}{t{i}}(\gamma{i}(s)) \, \mathrm{d}\nu(s)}{\leq \frac{\varepsilon}{3n} + c{t{i}}(x{i}, x{i+1}) \, \mathrm{by} \, (4.34)} + \underbrace{\sum{k=0}^{d} \left( \sum{i=i{\text{out}}}^{\sum{i=i{\text{out}}}^{k+1}} \int{\overline{S}{\text{exp}}^{i,k}} \underbrace{|\mathcal{L}{t{i}}(\gamma{i}(s)) - \mathcal{L}{t}(\gamma{i}(s))|}{\leq \varepsilon{\mathcal{L}} \, \mathrm{by} \, (4.33)} \right)}{\leq \frac{\varepsilon}{3} \, \mathrm{by} \, (4.37)} \ & \leq \frac{\varepsilon}{3} + \sum{i=1}^{n-1} c{t{i}}(x{i}, x{i+1}) + \frac{\varepsilon}{3} = \sum{i=1}^{n-1} c{t{i}}(x{i}, x_{i+1}) + \frac{2\varepsilon}{3}. \end{split}$$

Estimating ct(p k in, p^k out): We fix 0 ≤ k ≤ d. If k = 0, we only consider the case where p 0 in ̸= p 0 out, and we are thus in the first case of Equation (4.36), i.e., i k in, s^k in ∈ IScheap. If p 0 in = p 0 out, we may simply ignore this term in the triangular inequality at the end. We set j := J(i k in, s^k in) = J(i k out, s^k out). We can choose p k in, p k out ∈ C^j such that

$$d{\mathscr{P}}(\overline{p}{\mathrm{in}}^k, p{\mathrm{in}}^k) \leq r \leq L \quad \text{and} \quad d{\mathscr{P}}(\overline{p}{\mathrm{out}}^k, p{\mathrm{out}}^k) \leq r \leq L,$$

so that we can use Equation (4.31) to see that

$$\mathfrak{c}t(\overline{p}{\mathrm{in}}^k,p_{\mathrm{in}}^k) \leq \frac{\varepsilon}{6(m+1)} \quad \text{and} \quad \mathfrak{c}t(\overline{p}{\mathrm{out}}^k,p_{\mathrm{out}}^k) \leq \frac{\varepsilon}{6(m+1)}.$$

Using the triangular inequality and Property 3, we have that

$$\mathfrak{c}t(p{\mathrm{in}}^k,p_{\mathrm{out}}^k) \leq \mathfrak{c}t(p{\mathrm{in}}^k,\overline{p}_{\mathrm{in}}^k) + \mathfrak{c}t(\overline{p}{\mathrm{in}}^k,\overline{p}_{\mathrm{out}}^k) + \mathfrak{c}t(\overline{p}{\mathrm{out}}^k,p_{\mathrm{out}}^k) \leq \frac{\varepsilon}{3(m+1)}.$$

Conclusion: Since (i d+1 in , sd+1 in ) = (n, bn) and (i 0 in, s⁰ in) = (1, a1), we have that

$$c{t}(x{1}, x{n}) \leq \sum{k=0}^{d} c{t}(p{\text{in}}^{k}, p{\text{out}}^{k}) + \sum{k=0}^{d} c{t}(p{\text{out}}^{k}, p_{\text{in}}^{k+1})$$

$$\leq (m+1)\frac{\varepsilon}{3(m+1)} + \sum{i=1}^{n-1} c{t{i}}(x{i}, x{i+1}) + \frac{2\varepsilon}{3} = \sum{i=1}^{n-1} c{t{i}}(x{i}, x{i+1}) + \varepsilon.$$

The fact that for x, x′ ^∈ ^C := {^y ^∈ ^X | |∂tE^t |(y) = 0}, ct(x, x′ ) = 0 if and only if x and x ′ belong to the same path-connected components of C follows from Property 3 together with Property 2. □

To end this section, we proof the ancillary lemma Lemma 4.7.

Proof of Lemma 4.7. Let c^t be an action generated by curves as per Definition 3. We fix p1, p2, p³ ∈ P and ε > 0. We choose γ (i) ^∈ Γ(pⁱ , pi+1) for i ∈ {1, 2} such that

$$\mathbf{c}{t}(p{i}, p{i+1}) + \frac{\varepsilon}{2} \ge \int{(a{i}, b{i}]{\mathbb{I}}} \mathcal{L}{t}(\gamma^{(i)}(s)) \,\mathrm{d}\nu(s), \tag{4.38}$$

where ^aⁱ < bⁱ ^∈ ^I are such that Dom(^γ ) = [aⁱ , bⁱ . In particular, γ (ai) = xⁱ , γ (bi) = xi+1 and γ (1)(b1) = γ (2)(a2). We set γ (2) to be the curve γ (2) reparametrized to [b1, b² + b¹ − a2]^I , i.e, γ (2)(s) = γ (2)(^s ⁺ ^b¹ ⁻ ^a2). By Property 1, ^γ (2) ^∈ Γ(p2, p3) and likewise, the concatenation γ (1) ^∪ ^γ (2) as defined in Property 1 is in Γ(p1, p3). We have that

$$\begin{aligned} \mathbf{c}{t}(p{1}, p{3}) &\leq \int{(a{1}, b{2} + b{1} - a{2}]{\mathbb{I}}} \mathcal{L}{t}\left(\gamma^{(1)} \cup \underline{\gamma}^{(2)}(s)\right) d\nu(s) \ &\leq \int{(a{1}, b{1}]{\mathbb{I}}} \mathcal{L}{t}\left(\gamma^{(1)}(s)\right) d\nu(s) + \int{(a{2}, b{2}]{\mathbb{I}}} \mathcal{L}{t}\left(\gamma^{(2)}(s)\right) d\nu(s) \ &\leq \mathbf{c}{t}(p{1}, p{2}) + \mathbf{c}{t}(p{2}, p{3}) + \varepsilon. \end{aligned}$$

Since ε > 0 was arbitrary, we have that c^t fulfills the triangular inequality. The triangle inequality for c^t follows by definition of c^t in (4.10). □

We are now ready to state the main result of this paper, recalling the definition of X and ^q : [0, T] ^× ^X 7→ X from Equation (2.3).

Theorem 1 (complete). Let us assume Assumptions 1 to 5. Furthermore, let the family of mappings ω¯^t : ^X ^→ ^X indexed by ^t ^∈ [0, T] be the transition rule as in Definition 1, corresponding to an action c^t and complying with Axioms 1 to 3. Then, for any positive vanishing sequence (δn)n∈N, and for the corresponding discrete quasistatic evolutions η δn constructed according to Definition 1 we can—without relabeling—extract a subsequence such that:

(1) There exists a positive Radon measure ^µ¯ ∈ M([0, T]) and D ∈ ^L∞([0, T], ^R) such that
- µ δ ⁿ ⇀^∗ µ¯, where µ ^δⁿ is as defined in Equation (3.3), and
- ^Dδⁿ ⇀^∗ ^D in ^L∞([0, T], ^R), where ^Dδⁿ (t) := ^∂tE^t η ^δⁿ (t)
(2) The compositions q ◦ (id × η ^δⁿ ) converge pointwise to a piecewise continuous limiting curve ηˆ: [0, T] → X .
(3) The left and right limits ηˆ −(t) and ηˆ ⁺(t) of ^η^ˆ exist for every ^t ^∈ (0, T), and so do the limits ηˆ ⁺(0) and ηˆ −(T).
(4) The limiting curve ηˆ fulfills, for all 0 ≤ s < t ≤ T, the energy balance identiy

$$\hat{E}(\hat{\eta}^+(t)) - \hat{E}(\hat{\eta}^-(s)) = \int_s^t \mathcal{D}(\tau) d\tau - \bar{\mu}([s,t]), \tag{4.39}$$

(5) For all ^t ^∈ [0, T], ^ηˆ(t) ⊂ {(t, x) ^∈ [0, T] ^× ^X | |∂E^t |(x) = 0}.
(6) The limiting curve ηˆ is continuous in [0, T] \ J, where we define J := {t ∈ [0, T] | µ¯({t}) > 0}.

If we assume Assumption 6' in addition to Assumptions 1 to 5—here, Axioms 1 to 3 are sufficient—, we have that

(7) For almost all t ∈ [0, T], for all (t, x) ∈ ηˆ(t), we have that ∂tEt(x) = D(t). Picking any lifts h^t : X → [0, T], h^X : X → ^X such that ^q ◦ (h^t ^× ^hX) ^≡ id^X , we can thus rewrite the energy balance (4.39) as

$$\hat{E}(\hat{\eta}^{-}(s)) - \hat{E}(\hat{\eta}^{+}(t)) = \int{s}^{t} \partial{t} E{\tau} \left( h{\mathbb{X}} (\hat{\eta}(\tau)) \right) d\tau - \bar{\mu}([s, t]). \tag{4.40}$$

If the transition rule fulfills the stronger Axiom 3' in place of Axiom 3,—which is the case whenever we assume in addition that the transition rule is compatible with an action generated by curves—we have, furthermore,

(8) For every t ∈ J,

$$\hat{E}(\hat{\eta}^{-}(t)) - \hat{E}(\hat{\eta}^{+}(t)) = \bar{\mu}({t}) = \hat{c}_{t}(\hat{\eta}^{-}(t), \hat{\eta}^{+}(t)),$$

where cˆt([(t, x1], [(t, x2)]) := ct(x1, x2).

If we assume Assumptions 6 and 7 in addition to Assumptions 1 to 5 and 8—here, Axioms 1 to 3 are sufficient and Assumption 6' is not needed—, we even have that ∂tE· : [0, T] ^× ^X ^→ ^R factors through ^X ; we call the resulting functional ^∂tE^ˆ : X → ^R. In this case, we even have

(9) The limiting curve ηˆ fulfills, for all 0 ≤ s < t ≤ T, the energy balance identiy

$$\hat{E}(\hat{\eta}^{-}(s)) - \hat{E}(\hat{\eta}^{+}(t)) = \int{s}^{t} \partial{t} \hat{E}(\hat{\eta}(\tau)) d\tau - \bar{\mu}([s, t]). \tag{4.41}$$

Furthermore, we have that supp ¯µ = J, i.e., µ¯ is purely atomic.

Proof. Item 1 follows from Proposition 3.5 under Assumptions 1 to 5. Items 2 to 6 follows from Theorem 3.9 under Assumptions 1 to 5 whenever the action fulfills Axioms 1 to 3. Item 8 and that Axiom 3' implies Axiom 1 follows from Lemma 4.1 and Corollary 4.3 under Assumptions 1 to 5. The fact that Axiom 3' is guaranteed whenever we assume in addition that the transition rule is compatible with an action which is generated by curves is the content of Lemma 4.8.

Item 7 follows from Corollary 4.3 under Assumptions 1 to 5 whenever the action fulfills Axioms 1 to 3.

The fact that assuming Assumptions 6 to 8 in addition to Assumptions 1 to 5, we

can rewrite the energy balance as in Equation (4.41) follows from Lemma 3.10. In Proposition A.3, we show that under the same assumptions, we have that supp ¯µ = J. □

5. Examples of transition rules

The goal of this section is to show that the transition rules of the examples given in Section 4.2 are indeed actions which are generated by curves as per Definition 3, i.e., to prove Propositions 4.4 to 4.6. Before we dive into the specific examples, we first show a general lemma which will help us to prove Property 3 for all three examples.

Lemma 5.1. Assume Assumptions 4 and 5. Let c^t : ^X ^× ^X ^→ [0, ^∞) be an action which fulfills the triangular inequality, i.e., for which, for x, x′ , x′′ ^∈ ^X, we have that

$$c_t(x, x'') \le c_t(x, x') + c_t(x', x'').$$

Let us furthermore assume that there exists α > ⁰ such that for each compact ^K ^⊆ ^X there exists a constant ^C^K ^> ⁰ for which, for all x, x′ ^∈ ^K with x, x′ ∈ {^y ^∈ ^X ^| |∂E^t |(y) = 0} =: C , we have that ct(x, x′ ) ≤ C^K · d(x, x′ ) ^α. If x, x′ lie in the same component of C , and there exists some connected subset U ⊆ C such that x, x′ ⊆ U and U has Hausdorff dimension dimH(U) < α, then ct(x, x′ ) = 0.

Moreover, if we have that ^ct(y,y′ ^d(y,y′)^α → 0 as d(y, y′ ) → 0, then the same condition holds if the set U to have finite α-dimensional Hausdorff measure.

Proof. Let us pick x, x′ and U as in the assumptions. We can assume, without loss of generality, that U is contained in a single component of C , and thus, by Assumptions 4 and 5, ^U ^⊆ ^K for some compact ^K ^⊆ ^X. We pick ^C^K as in the assumptions. Next, we note that if U1, . . . , Uⁿ is an open cover of U, there exists—possibly after reordering—a sequence of distinct y1, . . . , yn′ ∈ U such that n ′ ≤ n + 1, {yⁱ , yi+1} ⊆ Uⁱ and y¹ = x, yn′ = x ′ . To see this, we first may assume without loss of generality that x ∈ U¹ and pick n ′ ^∈ ^N sucht that ^x ′ ∈ Un′. We consider the graph G whose vertices are given as N := {U1, . . . , Un} and whose edges are given by E := {(Uⁱ , U^j ) | Uⁱ ∩ U^j ̸= ∅}. Since U is connected, G is connected and we can find a minimal path from U¹ to Un′ in G. After reordering, we may assume that this path is given as U1, . . . , Un′. Setting y¹ = x, y n ′ = x ′ and picking y ⁱ ^∈ ^Uⁱ ^∩ ^Ui+1 for ⁱ ∈ {1, . . . , n′ ⁻ ¹}, we have found the desired sequence. By applying the triangular inequality repeatedly, we have that

$$ct(x,x') \leq \sum{i=1}^{n'-1} c_t(yi,y{i+1}) \leq \sup_{1 \leq j < n'} \frac{c_t(yj,y{j+1})}{d(yj,y{j+1})^{\alpha}} \sum_{i=1}^{n'-1} d(yi,y{i+1})^{\alpha}.$$

If either the Hausdorff dimension of ^U is smaller than ^α or if ^ct(y,y′ ^d(y,y′)^α → 0 as d(y, y′ ) → 0 and the α-dimensional Hausdorff measure of U is finite, we can make the right hand side arbitrarily small, proving the claim. □

5.1. The gradient flow. In this section, we prove Proposition 4.4. We use freely Equations (4.12) to (4.16) in Proposition 4.4. Substituting those definitions into Equa tion (4.10) and simplifying, we obtain the function c t : ^X ^× ^X ^→ [0, ^∞) defined in Equation (4.11).

Lemma 5.2. The map c F t coincides with c F ′ t : ^X ^× ^X ^→ [0, ^∞) defined as

$$c_t^{F'}(x_1, x2) := \inf{\phi \in AC([0,1]; \mathbb{X})} \int_{[0,1]} |\partial E_t|(\phi(s))|\dot{\phi}(s)| \,\mathrm{d}s.$$ (5.1)

Furthermore, the integral on the right-hand side is invariant to reparametrization of the curve: For all ^ϕ ^∈ AC([0, 1]; ^X) and all continuously differentiable and monotone increasing f : [a, b] → [0, 1], setting ϕ := ϕ ◦ f we have that for every t ∈ [0, T]

$$\int_{[0,1]} |\partial Et|(\phi(s))|\dot{\phi}(s)| \,\mathrm{d}s = \int{[a,b]} |\partial E_t|(\overline{\phi}(s))|\dot{\overline{\phi}}(s)| \,\mathrm{d}s. \tag{5.2}$$

As a result, we have that

$$c_t^F(x_1, x_2) := \inf \left{ \int_a^b |\partial E_t| \left( \phi(s) \right) \mathrm{d}s \, \left| \, \begin{array}{c} a < b \in \mathbb{R} \ \phi \in AC([a,b], \mathbb{X}) \ \phi(a) = x_1, \phi(b) = x_2, |\dot{\phi}| \equiv 1 \end{array} \right}. \tag{5.3}$$

Proof. Equation (5.2) is a simple application of the chain rule. For Equation (5.1), we note that for x, y ^∈ ^R, we have that

$$\frac{1}{2}x^2 + \frac{1}{2}y^2 = xy + \frac{1}{2}(x - y)^2 \ge xy.$$ (5.4)

From (5.4), it follows directly that c ^t ≥ c F ′ t . For the other direction, let us assume that c F t (x1, x2) ^> 0. We first fix some ^x1, x² ^∈ ^X ^× ^X and ^κ ^∈ (0, 3 ). Next, we choose some ε ′ ^> 0, b > 0, and a curve ^ϕ ^∈ AC([0, b], ^X) such that

$$\int_{[0,b]} |\partial E_t|(\phi(s))|\dot{\phi}(s)| \, \mathrm{d}s \le c_t^{F'}(x_1, x_2) + \varepsilon'. \tag{5.5}$$

By Equation (5.2), we can assume that ^|ϕ˙(s)| ≤ 1 for a.e. ^s ^∈ [0, b]. Let us fix ^ε ^∈ (0, 1) and set

$$r_{\varepsilon}(s) := \int_0^s \frac{|\dot{\phi}(\sigma)| + \varepsilon}{|\partial E_t|(\phi(\sigma)) + \varepsilon^{1+\kappa}} d\sigma \quad \text{for } s \in [0, b].$$

We define ϕ^ε := ϕ ◦ r −1 ^ε and note that D := Dom(ϕε) ⊆ [0, (M+1)b ε 1+^κ ]. If we set D′ := {s ∈ D | ^|ϕ˙(^r −1 ^ε (s))| |∂Et|(ϕ(r −1 ^ε (s))) ^≤ bεκ}, we see that for all ^σ ^∈ ^D′ , ˙rε(r −1 ε (σ)) ≤ b+1 ε ^κ , and thus |D′ | ≤ ε κ , where |D′ | denotes the Lebesgue measure of D′ . On the other hand, for s ∈ D \ D′ , we have—setting s ′ := r −1 ε (s)—that

$$\left| |\partial Et| (\phi{\varepsilon}(s)) - |\dot{\phi}_{\varepsilon}|(s) \right| = \left| |\partial E_t| (\phi(s')) - |\dot{\phi}|(s') \cdot \frac{|\partial E_t| (\phi(s')) + \varepsilon^{1+\kappa}}{|\dot{\phi}|(s') + \varepsilon} \right|$$

$$= \left| |\partial E_t| (\phi(s')) - \left(1 - \frac{\varepsilon}{|\dot{\phi}|(s') + \varepsilon}\right) (|\partial E_t| (\phi(s')) + \varepsilon^{1+\kappa}) \right|$$

$$\leq 2\varepsilon + \varepsilon \frac{|\partial E_t| (\phi(s'))}{|\dot{\phi}|(s')} \leq 2\varepsilon^{1+\kappa} + \frac{\varepsilon^{1-\kappa}}{b}.$$ (5.6)

Furthermore, we can find a bound N such that N ≥ |∂E^t ϕε(s) −|ϕ˙ ^ε|(s) for all s ∈ D, independendly of the chosen ^ε ^∈ (0, 1), as both ^|ϕ˙ | and |∂E^t | ϕ(s ′ ) are bounded. Using Equation (5.6) and Equation (5.4), we see that

$$c_t^F(x_1, x_2) \leq \frac{1}{2} \int_D |\partial Et|^2(\phi{\varepsilon}(s)) + |\dot{\phi}_{\varepsilon}(s)|^2 ds$$

$$= \int_D |\partial Et|(\phi{\varepsilon}(s)) |\dot{\phi}_{\varepsilon}(s)| ds + \int_D \frac{1}{2} (|\partial Et|(\phi{\varepsilon}(s)) - |\dot{\phi}_{\varepsilon}(s)|)^2 ds$$

$$\leq c_t^{F'}(x_1, x2) + \varepsilon' + \int{D \setminus D'} \frac{1}{2} \left( 2\varepsilon^{1+\kappa} + \frac{\varepsilon^{1-\kappa}}{b} \right)^2 ds + \int_{D'} \frac{1}{2} N^2 ds$$

$$\leq c_t^{F'}(x_1, x_2) + \varepsilon' + \frac{(M+1)b\varepsilon^{2-2\kappa}(2\varepsilon^{2\kappa} + \frac{1}{b})^2}{2\varepsilon^{1+\kappa}} + \frac{N^2 \varepsilon^{\kappa}}{2}.$$ (5.7)

As κ was chosen in (0, 3 ), we may pass to the limit in (5.7) as ε → 0, to obtain the inequality c F t (x1, x2) ≤ c F ′ t (x1, x2) + ε ′ . Letting ε ′ → 0 in a second step, we obtain the thesis. □

Lemma 5.3. Let us assume Assumptions 1 and 2. Let us consider a < b ^∈ ^R, ^x1, x² ^∈ ^X, and ^ϕ ^∈ AC([a, b]; ^X) such that ^ϕ(a) = ^x1, ^ϕ(b) = ^x2. Then

$$\int{a}^{b} \mathcal{L}{t}^{F}\left(\left(\phi(s), \dot{\phi}(s)\right)\right) ds \ge |E{t}(x{1}) - E{t}(x{2})|. \tag{5.8}$$

In particular, for any two points ^x1, x² ^∈ ^X and ^v1, v² ^∈ ^R ⁺, we have that

$$c_t^F((x_1, v_1), (x_2, v_2)) \ge |E_t(x_1) - E_t(x_2)|.$$ (5.9)

Proof. By Equation (5.4), we have that

$$\int_a^b \mathcal{L}_t^F\left(\left(\phi(s),\dot{\phi}(s)\right)\right)\mathrm{d}s \geq \int_a^b |\partial E_t|(\phi(s))|\dot{\phi}(s)|\,\mathrm{d}s.$$

Hence, Equation (5.8) follows from |∂E^t |(ϕ(s)) being a strong upper gradient of E^t as per Remark 1 (see [7, Definition 1.2.1]). Equation (5.9) follows by taking the infimum over all curves ϕ ∈ Γ (x1, v1),(x2, v2) in Equation (5.8). □

Before we show that c t is an action generated by curves, we will show an intermediate result.

Lemma 5.4. Let us assume Assumption 7, and let ^x ^∈ ^X and P, r ^∈ ^R such that |∂E^t |(y) ≤ P for all y ∈ Br(x). Then, for all x ′ ∈ Br(x), we have that

$$c_t^F(x, x') \le P \cdot d(x, x').$$

Proof. We pick ε > 0 arbitrary and, using Assumption 7, an ε-almost geodesic with natural parametrization γ between x and x ′ —where d(x, x′ ) + ε ≤ r—, i.e., |γ˙ | ≡ 1, γ(0) = x and γ(b) = x ′ , where b = d(x, x′ ) + ε. Since |∂E^t γ(s) ≤ P for s ∈ [0, b], we can follow c t x, x′ ) ≤ P d(x, x′ ) + ε directly by using Lemma 5.2. Since ε was arbitrary, the claim follows. □

Let us show that c t is actually an action.

Lemma 5.5. Let us assume Assumptions 1, 2 and 7. Then c t is an action according to Definition 2. In fact, c F t is even continuous.

Proof. To prove the continuity of c F t , we first note that in the spirit of Lemma 4.7, the triangular inequality follows from the fact that Γ is closed under restrictions, shifts and concatenations. Combining the triangular inequality with Lemma 5.4, we see that |ct(x1, x2) − ct(x ′ 1 , x′ 2 )| ≤ C1d(x1, x′ 1 ) + C2d(x2, x′ 2 ) for (x ′ 1 , x′ 2 ) close to (x1, x2), where C¹ = |∂E^t |(x1) + 1 and C² = |∂E^t |(x2) + 1. Finally, Equation (4.6), which requires that Et(x) − Et(x ′ ) ≤ ct(x, x′ ), follows as a special case of Equation (5.9). □

Before we prove that the action c t is generated by curves, we first investigate under which circumstances we can apply Lemma 5.1to c F t .

Lemma 5.6. Let us assume Assumptions 1, 2, 4, 5 and 7, and let x 7→ |∂E^t |(x) be locally α-H¨older continuous with α ∈ (0, 1]. Furthermore, assume that for each x, x′ which lie in the same component of ^C := {^x ^∈ ^X | |∂E^t |(x) = 0}, there exists some connected U ⊆ C such that x, x′ ∈ U and dimH(U) < 1 + α. Then, c F t (x, x′ ) = 0 whenever x, x′ lie in the same connected component of C .

Furthermore, if |∂E^t | is only continuous, we have that c F t (x, x′ ) = 0 whenever x, x′ lie in some connected U ⊆ C with finite 1-dimensional Hausdorff measure.

Proof. We start with the case α ∈ (0, 1). By the virtue of Lemma 5.1, we only need to show that for all compact ^K ^⊆ ^X, there exists a constant ^C^K ^> 0 such that for all x, x′ ∈ K ∩ C we have that ct(x, x′ ) ≤ C^K · d(x, x′ ) α+1 .

Let us fix a compact ^K ^⊂ ^X and let D > 0 be such that the ^α-H¨older inequality holds for all y, y′ ∈ K¹ := B1(K):

$$|\partial E_t|(y) - |\partial E_t|(y')| \le D \cdot d(y, y')^{\alpha}.$$

Let x, x′ ∈ C ∩ K. We set d := d(x, x′ ) and pick ε ∈ (0, 1) and a curve ϕ such that Dom(ϕ) = [0, d + ε], ϕ(0) = x, ϕ(d + ε) = x ′ and ^|ϕ˙ | ≡ 1. In particular, ϕ(t) ∈ K¹ for every t ∈ [0, d + ε]. We furthermore set the constants α := 1 − α and s ⋆ := d+ε α α , the

function m: [0, s^⋆ ] → [0, d + ε] defined as m(s) := (α · s) ^α , and ϕ := ϕ ◦ m. Then, for every ^s ^∈ [0, s^⋆ ] we have that

$$|\partial E_t|(\underline{\phi}(s)) \le D \cdot d(x,\underline{\phi}(s))^{\alpha} \le D \cdot m(s)^{\alpha} = D \cdot (\underline{\alpha} \cdot s)^{\frac{\alpha}{\alpha}},$$

and that

$$|\underline{\dot{\phi}}(s)| \equiv m'(s) = (\underline{\alpha} \cdot s)^{\frac{1}{\underline{\alpha}} - 1} = (\underline{\alpha} \cdot s)^{\frac{\underline{\alpha}}{\underline{\alpha}}}.$$

We now calculate

$$c^{F}(x,x') \leq \int{0}^{s^{\star}} |\partial E{t}|^{2}(\underline{\phi}(s)) + |\underline{\dot{\phi}}(s)|^{2} ds \leq \underbrace{(D^{2}+1) \cdot \underline{\alpha}^{\frac{2\alpha}{\underline{\alpha}}}}{=:D} \int{0}^{s^{\star}} s^{\frac{2\alpha}{\underline{\alpha}}} ds$$ $$= \frac{\overline{\alpha}D'}{1+\alpha} s^{\frac{1+\alpha}{\underline{\alpha}}} \Big|{0}^{s^{\star}} = \frac{\overline{\alpha}D'}{1+\alpha} (s^{\star})^{\frac{1+\alpha}{\underline{\alpha}}} = \underbrace{\frac{\overline{\alpha}D'}{1+\alpha} \cdot \frac{1}{\underline{\alpha}^{1+\alpha}}}{=:C_{K}} (\underline{d}+\varepsilon)^{1+\alpha}.$$

Since ε > 0 was arbitrary, we have that ct(x, x′ ) ≤ C^K · d(x, x′ ) 1+α.

For α = 1, we can do the same calculation as above, but setting m to be the function s 7→ e s instead.

To show the second part of the lemma, we again do a similar calculation as above with

m being the identity function to see that ct(y, y′ ) → 0 as d(y, y′ ) → 0 for y, y′ ∈ U. If we apply the second part of Lemma 5.1 with α = 0, the thesis follows. □

We are now ready to prove that the action c F t is generated by curves.

Lemma 5.7. Let us assume Assumptions 1, 2, 4, 5 and 7. Then, the action c F t is generated by curves as in Definition 3.

Proof. The continuity of L F ^t on P^F × [0, T] follows immediately from the continuity of |∂E^t ^| on ^X ^× [0, T]; likewise, the fact that ⁱ is a closed immersion is immediate. As for the properties, Properties 1 and 2 are immediate; Property 5 follows from Lemma 5.4: For ε > 0 and K ⊆ P^F compact, K^X := pX(K), the projection of K onto X, is compact as well. Due to the continuity of x 7→ |∂E^t |(x), we can find some r ′ , P such that for all x ∈ KX, x ′ ^∈ ^X for which ^d(x, x′ ) ≤ r ′ and |∂E^t |(x) ≤ P, we have that |∂E^t |(x ′ ) ≤ √ ε and we can set L := min(r ′ , √ ε). To finish, we note that c F t (p1, p2) does not depend on the second components of p1, p² ∈ P^F . Let us now prove the other two properties.

Property 3: To see that c F t (x1, x2) = 0 whenever x¹ and x¹ lie in the same connected component of the set of critical points ^C := {^x ^∈ ^X | |∂E^t |(x) = 0}, we note that x¹ and x² are connected by a rectifiable curve by Assumption 5. Since a rectifiable curve has finite 1-dimensional Hausdorff measure, we can apply Lemma 5.6 to see that c t (x1, x2) = 0. For the converse, let ^x1, x² ^∈ ^X such that they do not lie in the same connected component of K′ . Then either there is an i ∈ {1, 2} such that L F t (i F (xi)) > 0, or x¹ and x² lie in different connected components of C . In the former case, there exists an r ≤ d(x1, x2) such that P := minx∈Br(xi) |∂Et(x)| > 0 and, by reparametrizing any curve γ from x¹ to x² such that |γ˙ | ≡ 1 and using (5.2), we see that c F t (x1, x2) = c F ′ t (x1, x2) ≥ rP > 0. If x1, x² lie in different components of C , we set C¹ to be the component containing x¹ and use Assumption 5 to choose r such that |∂Et(x)| > 0 for all x ∈ Br(C1) \ C1. We set D := Br(C1) \ B^r (C1) and P := minx∈^D |∂Et(x)| > 0. Since any curve γ connecting x¹ and x² has to pass through D, we can use (5.2) again to reparametrize γ such that |γ˙ | = 1 and see that c F ′ t (x1, x2) ≥ r 2 P > 0.

Property 4: Fix some K ⊆ P^F compact. Then KX := pX(K), the projection of K onto X, is compact as well. Since c F t is continuous by Lemma 5.5, we can set C := maxx1,x2∈K^X c F t (x1, x2). We furthermore set Emax := maxx∈K^X Et(x). For each ε > 0 and ^p1, p² ^∈ ^K, we set ^x¹ := ^pX(p1) and ^x² := ^pX(p1) the projection onto ^X of ^p¹ and p2, respectively. With this notation, let us show that the first component of every quasi-optimal curve γ ∈ Γ(p1, p2), γ : [a, b] → P^F such that

$$\int{a}^{b} \mathcal{L}{t}^{F}(\gamma(s)) \, \mathrm{d}s \le c{t}^{F}(p{1}, p_{2}) + \varepsilon$$

stays within K′′ := E −1 t ((−∞, Z]) ^⊆ ^X, where ^Z := ^Emax ⁺ 1 2 (C + ε + 1). To see this, assume otherwise and pick s ^⋆ ^∈ (a, b) such that ^Et(γx(^s ⋆ )) > Z. Using Equation (5.8), we would then have the following contradiction:

$$\begin{split} C + \varepsilon &\geq \int{a}^{b} \mathcal{L}{t}^{\scriptscriptstyle{F}}(\gamma(s)) \, \mathrm{d}s = \int{a}^{s^{\star}} \mathcal{L}{t}^{\scriptscriptstyle{F}}(\gamma(s)) \, \mathrm{d}s + \int{s^{\star}}^{b} \mathcal{L}{t}^{\scriptscriptstyle{F}}(\gamma(s)) \, \mathrm{d}s \ &> (Z - E{max}) + (Z - E{max}) = 2Z - 2E_{max} = C + \varepsilon + 1. \end{split}$$

By Assumption 4, K′′ is compact. What is left to show is that we can choose quasioptimal curves with bounded velocity. To this end, note that in the proof of Lemma 5.2, we chose near-optimal curves γ whose velocity fulfilled

$$||\dot{\gamma}(s)| - |\partial E_t|(\gamma(s))| \le \varepsilon'$$

for arbitrarily ε ′ > 0. Thus we can set vmax := maxx∈K′ |∂E^t |(x) + ε ′ , for any ε ′ > 0, and K′ := K′′ × [0, vmax] ⊆ P^F . □

To finish this section, we prove Proposition 4.4.

Proof of Proposition 4.4. The fact that c t is an action as in Definition 2 is the content of Lemma 5.5; the fact that c t is generated by curves as in Definition 3 is the content of the above Lemma 5.7. What is left to prove is that ¯ω F ^t as defined in Definition 4 is compatible with c ^t as in Definition 2, i.e., that for all ^x1, x² ^∈ ^X and ^ϕ ^∈ AC([0, ^∞), ^X) such that ϕ(0) = x1, lims→∞ ϕ(s) = x² and, for all s ∈ [0, ∞),

$$E_t(x) - E_t(\phi(s)) = \frac{1}{2} \int_0^s |\partial E_t|^2 (\phi(s)) + |\dot{\phi}(s)|^2 ds,$$

we have that

$$E_t(x_1) - E_t(x_2) = c_t^F(x_1, x_2).$$

This follows directly from the continuity of E^t and c t . □

5.2. The minimizing movement scheme. In this section, we will explore the minimizing movement scheme and prove Proposition 4.5. We use freely the definitions Equations (4.19) to (4.23), and in the rest of this chapter adopt the convention where we denote the components of a curve ^γ : ^I ^⊆ ^Z ^→ ^P^M ⁼ ^X ^× ^X by ^γ⁰ and ^γ1, and we denote the components of points x, x′ ∈ P^M , by (x0, x1) := x and (x 0 , x′ 1 ) := x ′ . Substituting Equations (4.19) to (4.22) into Equation (4.10) and simplifying, we obtain the expression of ^τ c M t : ^X ^× ^X ^→ [0, ^∞) written in Equation (4.17). As we describe in Remark 7, whenever we assume that Assumption 7 holds, we also assume that τ ≤ L , where L is the Lipschitz constant mentioned in Assumption 7. Finally, we report that in [25, Definition 5.14] the authors introduced an action for the minimizing movement scheme in Euclidean setting which is fully compatible with ^τ c M t .

Remark 18. If we assume Assumptions 1 to 4, then the function (t, x, y) 7→ Et(y) + 1 2τ d(x, y) 2 is continuous in (t, x, y). Furthermore, only points in the relatively compact sublevel set {^y ^∈ ^X ^| ^Et(y) ^≤ ^Et(x)} contribute to the infimum in (4.18). Thus, ^E M t,τ is itself continuous in (t, x), as it is locally an infimum of equicontinuous functions. Hence, also ^τ^L M t is continuous in (t, x) ∈ [0, T] × P^M . Lastly, we note that E M t,τ (x) ≤ Et(x) for all ^x ^∈ ^X: To see this, we use ^y ⁼ ^x as a competitor in Equation (4.18). We clarify the relation between the conditions E t,τ (x) = Et(x) and |∂E^t ^|(x) = 0 for ^x ^∈ ^X in Lemma 5.13.

Let us first see how this definition relates to the minimizing movement scheme.

Lemma 5.8. Let us assume Assumptions 1 to 4. Then, for all τ > 0, x, x′ ∈ P^M and γ ∈ Γ(x, x′ ), we have that

$$\sum{s=a+1}^{b} \mathcal{T}{t}^{M} (\gamma(s)) \ge E{t}(x{1}) - E{t}(x{1}'), \tag{5.10}$$

If x = i ^M (x) and x ′ = i ^M (x ′ ) for some x, x′ ^∈ ^X, then

$$\sum{s=a+1}^{b} \mathcal{I}{t}^{M} (\gamma(s)) \ge E{t}(x) - E{t,\tau}^{M}(x'), \tag{5.11}$$

where equality is attained if and only if

$$\gamma_0(s+1) = \gamma_1(s) \in \underset{y \in \mathbb{X}}{\operatorname{arg min}} \left{ E_t(y) + \frac{1}{2\tau} d(\gamma_0(s), y)^2 \right}$$ (5.12)

for all ^s ^∈ [^a + 1, b ⁻ 1]Z. In particular, for every x, x′ ^∈ ^X we have

$$\tau_{c_t}^M(x, x') \ge Et(x) - E{t, \tau}^M(x') \ge E_t(x) - E_t(x').$$ (5.13)

Proof. To prove Equation (5.10), it suffices to show that ^τ^L M t γ(s) ≥ E^t γ0(s) − Et γ1(s) for all s ∈ [a + 1, b − 1]Z. To this end, we write

$$\mathcal{L}{t}^{M}(\gamma(s)) - \left(E{t}(\gamma{0}(s)) - E{t}(\gamma{1}(s))\right) \ = E{t}(\gamma{0}(s)) - \inf{y \in \mathbb{X}} \left{E{t}(y) + \frac{1}{2\tau}d(\gamma{0}(s), y)^{2}\right} \

\frac{1}{2\tau}d(\gamma{1}(s), \gamma{0}(s))^{2} - E{t}(\gamma{0}(s)) + E{t}(\gamma{1}(s)) \ \geq E{t}(\gamma{0}(s)) - E{t}(\gamma{1}(s)) - \frac{1}{2\tau}d(\gamma{1}(s), \gamma{0}(s))^{2} \
\frac{1}{2\tau}d(\gamma{1}(s), \gamma{0}(s))^{2} - E{t}(\gamma{0}(s)) + E{t}(\gamma{1}(s)) = 0,$$ (5.14)

where the inequality follows by using y = γ1(s) as a competitor for the infimum. Equa tion (5.11) follows by applying Equation (5.10) to ^γ|[a,b−1] and noting that ^τ^L M t γ(b) = τL M t (x ′ , x′ ) = Et(x ′ ) − E M t,τ (x ′ ). In the lower bound Equation (5.11), equality is attained if and only if γ0(s) fulfills (5.12). Finally, Equation (5.13) follows by taking the infimum over Γ(i ^M (x), i^M (x ′ )) in Equation (5.11) and using Remark 18 for the second inequality. □

Before we can show that ^τ c M t is an action according to Definition 2, we need to show two intermediate results. The first lemma is related to Property 5, the second lemma concerns Property 4 of Definition 3.

Lemma 5.9. Let us assume Assumptions 1 to 4. For all x, x′ ∈ P^M and L, P > 0 such that dP^M (x, x′ ) ≤ L, τL M t (x) ≤ P, τL M t (x ′ ) ^≤ ^P and ^τ^L M t (i ^M (x1)) ≤ P we have that τ c M t (x, x′ ) ≤ 2τ (2L + √ 2τP) ² + 3P.

Proof. Using Remark 18, we see that

$$\frac{1}{2\tau}d(x_0, x_1)^2 \le E_t(x0) - E{t,\tau}^M(x_0) + \frac{1}{2\tau}d(x_0, x_1)^2 = \mathcal{L}_t^M(\underline{x}) \le P, \tag{5.15}$$

Combining this with d(x0, x′ 0 ) ≤ 2dP^M (x, x′ ) ≤ 2L, which follows directly from the definition of dP^M in Equation (4.19), we see —using the triangular inequality— that d(x1, x′ 0 ) ≤ 2L + √ 2τP. Furthermore, we have for x ′′ := (x1, x′ 0 ), that ^τ^L M t (x ′′) ≤ 2τ d(x1, x′ 0 ) ² ⁺ ^P, since ^τ^L M t (i ^M (x1)) ≤ P. The lemma now follows by considering as a competitor for ^τ c M t the curve γ : [1, 3]^Z → P^M defined as γ(1) := x, γ(2) := x ′′ , γ(3) := x ′ and noting that, owing to the hypotheses, P³ s=2 τL M t (γ(s)) ≤ 1 2τ (2L + √ 2τP) ² + 3P. □

Lemma 5.10. Let us assume Assumptions 1 to 4. For all K ⊆ P^M compact and C > 0, there exists K′ ⊆ P^M compact such that the following implication holds for all x, x′ ∈ K and γ ∈ Γ(x, x′ ):

$$\sum{s=a+1}^{b} {}^{\mathsf{T}} \mathcal{L}{t}^{M}(\gamma(s)) \leq C \implies \operatorname{Im}(\gamma) \subseteq K'. \tag{5.16}$$

In particular, for each K ⊆ P^M there exists a compact set K ⊆ P^M such that for every ε ∈ (0, 1) the following implication holds for all x, x′ ∈ K and γ ∈ Γ(x, x′ ):

$$\sum{s=a+1}^{b} \mathcal{I}{t}^{M}(\gamma(s)) \leq \tilde{\mathfrak{c}}_{t}^{M}(\underline{x},\underline{x}') + \varepsilon \implies \operatorname{Im}(\gamma) \subseteq K'. \tag{5.17}$$

Proof. We start with (5.16) and fix ^K ^⊆ ^P^M and C > 0. Since Im(γ1|[a,b−1]) = Im(γ0|[a+1,b] ) for all γ ∈ Γ(x, x′ ), it suffices to show that Im(γ1) is compact in X. To proceed, we use the coercivity of E^t (cf. Assumption 4) and Equation (5.10) to get the following chain of inequalities, for any s ^⋆ ^∈ [^a + 1, b ⁻ 1]Z:

$$C \geq \sum_{s=a+1}^{b} {^{\mathsf{T}}!\mathcal{L}t^{M}} \left( \gamma(s) \right) \geq \sum{s=s^\star+1}^{b} {^{\mathsf{T}}!\mathcal{L}_t^{M}} \left( \gamma(s) \right) \geq E_t \left( \gamma_1(s^\star) \right) - E_t(x_1')$$

$$\geq E_t \left( \gamma1(s^\star) \right) - \max{(y_0, y_1) \in K} E_t(y_1).$$

To show (5.17), we infer from Lemma 5.9, from the continuity of ^τ^L M ^t —see Remark 18 and from the compactness of K that ^τ c t is bounded on K × K by some C ′ . Applying (5.16) with C = C ′ + 1 finishes the proof. □

We are now in a position to prove that the map ^τ c M t : ^X ^× ^X ^→ [0, ^∞) is an action.

Lemma 5.11. Let us assume Assumptions 1 to 4. Then, ^τ c M t is an action according to Definition 2.

Proof. We start by choosing an arbitrary compact subset ^K ^∈ ^X. It suffices to show that ^τ c t is lower semicontinuous on K × K. To this end, we show the continuity of τ c t : ^X ^× ^X ^→ [0, ^∞) on ^K, which is defined as follows:

$$\underline{\mathbf{T}}{t}^{M}(x,x') \coloneqq \inf \left{ \sum{s=a+1}^{b} \underline{\mathbf{T}}_{t}^{M}\left(\gamma(s)\right) \,\middle|\, \gamma \in \underline{\Gamma}\big(i^{M}(x),i^{M}(x')\big) \right}$$

where Γ(p, p′ ) = {γ ∈ Γ(p, p′ ) | Dom(γ) = [a, b]Z, a < b − 1}. Beside the condition a < b−1, the definition of ^τ c M t coincides with the definition of ^τ c M t in (4.17). The continuity of ^τ c M ^t on ^K ^×^K implies the lower semicontinuity of ^τ c M ^t on K ×K. Indeed, this follows from the facts that (1.) ^τ c M t ≥ τ c M ^t and (2.) the set of points where ^τ c M ^t ̸= τ c M t is the diagonal in ^K×K, which is closed —and on this diagonal, ^τ c M t (x, x) =τ^L M t (x, x) is continuous. For any a < b−¹ ^∈ ^Z, x, x′ ^∈ ^K and ^f : [a+1, b−1]^Z ^→ ^X, we set ^F(f, x, x′ ): [a, b]^Z → P^M to be the finite sequence

$$(i^{M}(x), (x, f(a+1)), (f(a+1), f(a+2)), \dots, (f(b), x'), i^{M}(x')).$$ (5.18)

Note here that we allow the case a = b − 2, in which case the above display should be read as i ^M (x),(x, x′ ), i^M (x ′ ) . Setting ^D := {^f : [^a + 1, b ⁻ 1]^Z ^→ ^X ^| a < b ⁻ ¹}, we observe that F(·, x, x′ ) is a bijection from D to Γ i ^M (x), i^M (x ′ ) . Thus, we can write τ c ^t as the infimum over such extensions:

$$\underline{\underline{\tau}}{\underline{t}}^{M}(x, x') = \inf \left{ \sum{s=a+1}^{b} \underline{\mathcal{T}}{\underline{t}}^{M} \left( F(f, x, x') \right) \middle| f \in D, \operatorname{Dom}(f) = [a+1, b-1]{\mathbb{Z}} \right}.$$ (5.19)

Furthermore, by the virtue of Lemma 5.10, we can take the infimum over only those f for which Im(f) ⊆ K′ for some compact K′ ⊆ P^M . Setting D′ := {f : [a + 1, b − 1]^Z → K′ ^| a < b ⁻ ¹} and defining the function ^C: ^D′ ^× ^K ^× ^K ^→ ^R by

$$\mathfrak{C}((f,x,x)) \coloneqq \sum_{s \in \text{Dom}(F(f,x,x))} \mathcal{T}_t^M (F(f,x,x')(s)).$$

We can thus rewrite Equation (5.19) as

$$\underline{\mathcal{C}}{\underline{t}}^{M}(x, x') = \inf{f \in D'} \left{ \mathfrak{C}(f, x, x') \right}. \tag{5.20}$$

For each f ∈ D′ , ^C(f, ·, ·) is continuous on ^K ^× ^K, due to the continuity of ^τ^L M ^t —see Remark 18. If we can show that this continuity is uniform over f ∈ D′ , then ^τ c M t is continuous on K × K, as it is the infimum over equicontinuous functions. To this end, we pick x, x′ , x, x ′ ∈ K, f ∈ D′ with Dom(f) = [a + 1, b − 1]^Z and use the definition of F in Equation (5.18) and write

$$\begin{split} & \mathfrak{C}(f,x,x') - \mathfrak{C}(f,\overline{x},\overline{x}') \ = & \left( \overline{\mathcal{L}}_t^M \left( i^M(x) \right) - \overline{\mathcal{L}}_t^M \left( i^M(\overline{x}) \right) \right) + \left( \overline{\mathcal{L}}_t^M \left( \left( x, f(a+1) \right) \right) - \overline{\mathcal{L}}_t^M \left( \left( \overline{x}, f(a+1) \right) \right) \right) \ & + \left( \overline{\mathcal{L}}_t^M \left( \left( f(b), x' \right) \right) - \overline{\mathcal{L}}_t^M \left( \left( f(b), \overline{x}' \right) \right) \right) + \left( \overline{\mathcal{L}}_t^M \left( i^M(x') \right) - \overline{\mathcal{L}}_t^M \left( i^M(\overline{x}') \right) \right). \end{split}$$

All the above terms go to zero as (x, x′ ) → (x, x ′ ), uniformly over f(a + 1), f(b) ∈ K′ , which shows the claimed equicontinuity. □

Our next goal is to show that the action ^τ c t is generated by curves. To this end, we again need to show two intermediate results.

Lemma 5.12. Let us assume Assumptions 1 to 4, and let x, x′ ∈ P^M such that there exists a compact connected set U ⊆ P^M for which dimH(U) < 2, {x, x′} ⊆ U and τL M t ^|^U ^≡ ⁰. Then ^τ c M t (x, x′ ) = 0.

Proof. Since ^τ^L M t |^U ≡ 0 implies that U ⊆ Im(i), we can apply Lemma 5.1 with α = 2 by using Lemma 5.9. □

Lemma 5.13. Let us assume Assumptions 1 to 4 and 7, and let x, x^⋆ ^∈ ^X and R > ⁰ such that {x ′ ^∈ ^X ^| ^Et(^x ′ ) ≤ Et(x)} ⊆ B ^R 4 (x ⋆ ). Then, we have that

$$|\partial E_t|(x) = 0 \iff \mathcal{L}_t^M(i^M(x)) = 0.$$

Proof. We start proving the implication ^τ^L M t (i ^M (x)) =⇒ |∂E^t |(x) = 0. If |∂E^t |(x) > 0, then there exists a sequence (x ′ n )n∈^N → x such that limn→∞ Et(x ⁿ)−Et(x) d(x′ ⁿ,x) > 0. We thus pick ε > 0 and a subsequence without relabeling such that ^Et(^x ⁿ)−Et(x) d(x′ ⁿ,x) ≥ ε for all n. For such a sequence, we have, for n large enough, that

$$\mathcal{T}{t}^{M}\left(i^{M}\left(x\right)\right) \geq E{t}(x) - E{t}(x'{n}) - \frac{d(x, x'{n})^{2}}{2\tau} \geq d(x, x'{n})\left(\varepsilon - \frac{d(x, x'_{n})}{2\tau}\right) > 0.$$

For the opposite implication, we assume that |∂E^t |(x) = 0. Then, for all y ̸= x for which Et(y) < Et(x), we set d := d(x, y) and use Assumption 7 to pick some 0 < ε ≤ 2 and some ε-geodesic γ between x and y with natural parametrization, i.e., such that γ(0) = x, γ(d + ε) = y and |γ˙ | ≡ 1 a.e. Since x, y ∈ B ^R 4 (x ⋆ ) by hypothesis, we note that Im(γ) ⊆ BR(x ⋆ ) and observe that by Remark 1 and Assumption 7,

$$\begin{split} E_t(x) - E_t(y) - \frac{d(x,y)^2}{2\tau} & \leq \left| \int_0^{\underline{d}+\varepsilon} |\partial E_t|(\gamma(s)) \, \mathrm{d}s \right| - \frac{\underline{d}^2}{2\tau} \leq \int_0^{\underline{d}+\varepsilon} L \cdot s \, \, \mathrm{d}s - \frac{\underline{d}^2}{2\tau} \ & \leq \frac{1}{2} L(\underline{d}+\varepsilon)^2 - \frac{\underline{d}^2}{2\tau} \leq \frac{\underline{d}^2}{2} \left( L - \frac{1}{\tau} \right) + \varepsilon L \left( \underline{d} + \frac{\varepsilon}{2} \right) \leq \varepsilon L \left( \underline{d} + \frac{\varepsilon}{2} \right). \end{split}$$

Since ε > 0 was arbitrary, we have that Et(x)−Et(y)− d(x,y) ²^τ ≤ 0. Since y was arbitrary as well, we have shown that ^τ^L M t (i ^M (x)) = 0 for such x. □

We are now ready to prove that the action ^τ c M t is generated by curves.

Lemma 5.14. Let us assume Assumptions 1 to 5 and 7. Then, for all τ ≤ 1 L , the action τ c M t is generated by curves as in Definition 3.

Proof. The fact that i ^M is a closed immersion is immediate, and the continuity of ^τ^L M t was shown in Remark 18. As for the properties, Property 1 is immediate, Property 2 results from Lemma 5.13, Property 4 follows directly from Lemma 5.10 and Property 5 descends from Lemma 5.9. We will now show the missing Property 3.

The assumptions and Lemma 5.12 imply that ^τ c t (x, x′ ) = 0 whenever i ^M (x) and i ^M (x ′ ) lie in the same connected component of i ^M (^C ) ^⊆ ^P^M , where ^C := {^x ^∈ ^X | |∂E^t |(x) = 0}. In this case, by Assumption 5 we have that x and x ′ are connected by a curve γ in X whose image in P^M under i ^M has Hausdorff dimension smaller than 2. Thus, we can apply Lemma 5.12. For the converse, we assume that i ^M (x) and i ^M (x ′ ) are not in the same connected component of i ^M (C )—because i ^M is an isometry, this is equivalent to x and x ′ not being in the same connected component of C . We choose ε > 0, K′ ⊆ P^M compact such that implication (5.17) in Lemma 5.10 holds for K = {i ^M (x), i^M (x ′ )} and define the compact set K := K′ ⁰ ∪ K′ 1 , where K′ 0 , K′ ¹ ^⊆ ^X are the projections of ^K′ over ^X. We set ^C1, . . . , ^Cⁿ to be the connected components of ^C which intersect ^K. We furthermore set r := mini̸=^j miny∈Ci,y′∈C^j d(y, y′ ), which is nonzero by Assumption 5. If x ′ ̸∈ C , we immediately see that ct(x, x′ ) ≥ L^t i ^M (x ′ ) > 0. So we may assume, after reordering, that x ′ ^∈ ^C¹ and ^x ̸∈ ^C1. We set ^D := K \ ^Sⁿ ⁱ=1 B^r 4 (Ci) and Pmin := miny∈^D Lt(i ^M (y)), which is positive by compactness of ^K and continuity of ^τ^L M t . Now pick any γ ∈ Γ i ^M (x), i^M (x ′ ) such that

$$\sum{s=a+1}^{b} \mathcal{I}{t}^{M}(\gamma(s)) \leq \tilde{c}_{t}^{M}(x, x') + \varepsilon$$ (5.21)

where a < b ^∈ ^Z are such that Dom(γ) = [a, b]Z. Let ^s ^⋆ be the maximal s such that γ0(s) ∈ B^r 4 (C1). Then γ1(s ⋆ ) ̸∈ B^r 4 (C1), and either γ1(s ⋆ ) ∈ D or γ1(s ⋆ ) ∈ B^r 4 (Ci) for some i ̸= 1. In the former case, we have that

$$\sum{s=a}^{b} \mathcal{I}{t}^{M}\left(\gamma(s)\right) \geq \mathcal{I}{t}^{M}\left(\gamma(s^{\star})\right) \geq \mathcal{I}{t}^{M}\left(i^{M}\left(\gamma{1}(s^{\star})\right)\right) \geq P{\min}.$$

In the latter case, we have that

$$\sum_{s=a}^b \mathcal{T}_t^{\scriptscriptstyle M}\left(\gamma(s)\right) \geq \frac{1}{2\tau} d!\left(\gamma_0(s^\star), \gamma_1(s^\star)\right)^2 \geq \frac{r^2}{8\tau}.$$

Since (5.21) holds and ε > 0 is arbitrary, we infer from the last two inequalities that τ c M t (x1, x2) ^≥ min( ^r 8τ , Pmin) > 0. □

To finish this section, we prove Proposition 4.5.

Proof of Proposition 4.5. Let τ > 0. The fact that ^τ c M t is an action as in Definition 2 is the content of Lemma 5.11; the fact that for |∂E^t | Lipschitz continuous with Lipschitz constant L and τ ≤ 1 L , τ c M t is generated by curves as in Definition 3 is the content of the above Lemma 5.14. What is left to prove is that ^τ ω¯ ^t as defined in Definition 5 is compatible with ^τ c ^t as in Definition 2, i.e., that for all x, x′ ^∈ ^X, (u^j )j∈^N ^∈ ^X and increasing sequences (s^j )j∈^N ^∈ ^N such that ^u⁰ ⁼ ^x, limj→∞ ^us^j ⁼ ^x ′ and

$$u_{j+1} \in \underset{y \in \mathbb{X}}{\operatorname{arg min}} \left{ E_t(y) + \frac{1}{2\tau} d(u_j, y)^2 \right}$$ (5.22)

for all s ∈ [0, ∞)Z, we have that

$$E_t(x) - E_t(x') = \overset{\tau}{c}_t^M(x, x').$$

However, this follows directly by applying Lemma 5.8 to the finite curves

$$(i(x), (u_0, u1), \dots, (u{sj-1}, u{sj}), i^M(u{sj})) \in \Gamma(i^M(x), i^M(u{s_j})),$$

using continuity of E^t and ^τ c M ^t and noticing that the convergence of Et(u^j ) implies that Et(usⁱ ) − E M t,τ (usⁱ ) ^→ 0 as ^s → ∞. □ 5.3. The BDF2 method for the gradient flow. In this section, we will explore the BDF2 discretization of the gradient flow and prove Proposition 4.6. To the best of our knowledge, we derive here for the first time the action related to the BDF2 scheme. We use freely the notation in Equations (4.26) to (4.30). In the rest of this section we denote the components of a curve ^γ : ^I ^⊆ ^Z ^→ ^P^B ⁼ ^X ^× ^X ^× ^X by ^γ−1, ^γ⁰ and ^γ1, and the components of points x, x′ ∈ P^B by (x−1, x0, x1) := x and (x ′ −1 , x′ 0 , x′ 1 ) := x ′ . As we described in Remark 7, whenever we assume that Assumption 7 holds, we also assume that τ ≤ L , where L is the Lipschitz constant mentioned in Assumption 7. Substituting Equations (4.26) to (4.29) into Equation (4.10) and simplifying, we obtain the action Equation (4.24).

Remark 19. In the definitions concerning the minimizing movement scheme in Equa tions (4.18) and (4.22), ^τ^L t resembled E t,τ . In contrast, in the definitions in Equa tions (4.25) and (4.29), ^τ^L B ^t and E B t,τ look quite different, with differing fractions involving τ . Based on what we have seen on the minimizing movement scheme, one might instead expect the following definition instead of Equation (4.29):

$$\widetilde{\mathcal{TL}{t}^{B}}(\underline{x}) := E{t}(x{0}) - E{t,\tau}^{B}(x{0}, x{-1}) + \frac{1}{\tau}d(x{0}, x{1})^{2} - \frac{1}{4\tau}d(x{-1}, x{1})^{2}.$$ (5.23)

In fact, we could have chosen the definition of ^τLf^B t in (5.23) in place of ^τ^L B t in (4.29): For every x, x′ ^∈ ^X, every (^γ ^∈ Γ(ⁱ B (x), i^B (x ′ )) with γ : [a, b]^Z → P^B , and every τ > 0, we can use the fact that d γ0(s), γ1(s) = d γ−1(s + 1), γ0(s + 1) for s ∈ [a + 1, b − 1]Z, while d γ−1(a), γ0(a) = d γ0(a), γ1(a) = 0 —and likewise for γ(b)— to see that

$$\sum{s=a+1}^{b} \mathcal{I}{t}^{B}(\gamma(s)) = \sum{s=a+1}^{b} \widetilde{\mathcal{I}{t}^{B}}(\gamma(s)). \tag{5.24}$$

On the other hand, our formulation has the advantage that ^τ^L B t is nonnegative, as we show in the next lemma.

Lemma 5.15. For any x ∈ P^B , we have that

$$\mathcal{L}{t}^{B}(\underline{x}) \ge E{t}(x{0}) - E{t,\tau}^{B}(x{0}, x{-1}),$$ (5.25)

where equality can only be attained if d(x−1, x0) = d(x0, x1). Furthermore, we have that

$$E_t(x0) - E{t,\tau}^B(x0, x{-1}) \ge \frac{1}{4\tau} d(x0, x{-1})^2 \ge 0.$$ (5.26)

Proof. The inequality (5.25) follows directly from the definition of ^τ^L B t in Equation (4.29) and the fact that, for x, x′ , x′′ ^∈ ^X, setting ^d¹ := d(x, x′ ) and d² := d(x ′ , x′′):

$$\frac{1}{4}d(x,x'')^2 \le \frac{1}{4}(\underline{d}_1 + \underline{d}_2)^2 \le \frac{1}{2}(\underline{d}_1^2 + \underline{d}_2^2),\tag{5.27}$$

where the first bound comes from the triangular inequality and the second bound is attained if d¹ = d² . We get inequality (5.26) using x⁰ as a competitor for the infimum in the definition of E t,τ in Equation (4.25). □

Using the reformulation Equation (5.24), we see that the BDF2 discretization of the gradient flows is indeed compatible with ^τ c t .

Lemma 5.16. For any x in P^B , we have that

$$\widetilde{\mathcal{L}_t^B}(\underline{x}) \ge E_t(x_0) - E_t(x_1),$$ (5.28)

where equality is attained if and only if x = (x−1, x0, x1) satisfies

$$x_1 \in \underset{y \in \mathbb{X}}{\operatorname{arg\,min}} \left{ E_t(y) + \frac{1}{\tau} d(y, x0)^2 - \frac{1}{4\tau} d(y, x{-1})^2 \right}.$$ (5.29)

In particular, for any x, x′ ^∈ ^X, we have that

τ c B t (x, x′ ) ≥ Et(x) − E B t,τ (x ′ , x′ ) ≥ Et(x) − Et(x ′ ). (5.30)

Proof. To show (5.28), we start with the following calculation: for every x ∈ P^B

$$\begin{split} \widetilde{\mathcal{L}}_t^B(\underline{x}) &= E_t(x0) - E{t,\tau}^B(x0, x{-1}) + \frac{1}{\tau} d(x_1, x_0)^2 - \frac{1}{4\tau} d(x1, x{-1})^2 \ &= E_t(x0) - \inf{y \in \mathbb{X}} \left{ E_t(y) + \frac{1}{\tau} d(y, x0)^2 - \frac{1}{4\tau} d(y, x{-1})^2 \right} \ &\quad + \frac{1}{\tau} d(x_1, x_0)^2 - \frac{1}{4\tau} d(x1, x{-1})^2 \ &\geq E_t(x_0) - E_t(x_1). \end{split}$$

In particular, equality is attained if and only if the infimum is achieved at x1, i.e., if x fulfills Equation (5.29). To show Equation (5.30), we first recursively apply Equation (5.28) to any γ ∈ Γ(x, x′ ) for x, x′ ∈ P^B to see that

$$\sum{s=a+1}^{b} \widetilde{\mathcal{L}{t}^{B}} (\gamma(s)) \ge E{t} (\gamma{1}(a)) - E{t} (\gamma{1}(b)).$$

Equation (5.30) then follows from the definition of ^τ c t , Equation (5.24) and the fact that i (x1)¹ = x¹ and i (x2)¹ = x2. □

Next, we investigate the relation between ^τ c B ^t and ^τ c M t . To avoid confusion, we denote the admissible curves for the BDF2 method, defined in Equation (4.28), by Γ^B , and we denote the admissible curves for the minimizing movement scheme, defined in Equa tion (4.21), by Γ^M . We start by defining a bijection between certain sets of admissible curves. For x, x′ ∈ P^B , we define F : Γ^B (x, x′ ) → Γ M (x0, x1),(x ′ −1 , x′ 0 ) as follows:

$$F(\gamma) := (\gamma_0(a), \gamma_1(a)), \dots, (\gamma_0(b-1), \gamma_1(b-1)),$$ (5.31)

where we choose the domain of ^F(γ) to be [a, b ⁻ 1]^Z and where a < b ^∈ ^Z are chosen such that Dom(γ) = [a, b]Z. From the definitions of Γ^B and Γ^M , both injectivity and surjectivity of F follow right away.

Lemma 5.17. For every τ > 0, every x, x′ ∈ P^B , and every γ ∈ Γ (x, x′ ) we have that

$$\frac{1}{5}\sum_{s=a+1}^{b-1} {}^{\frac{\tau}{2}} \mathcal{L}t^{\scriptscriptstyle M}\left(F(\gamma)(s)\right) \leq \sum{s=a+1}^b {}^{\tau} \mathcal{L}t^{\scriptscriptstyle B}\left(\gamma(s)\right) \leq 3\sum{s=a+1}^{b-1} {}^{\tau} \mathcal{L}_t^{\scriptscriptstyle M}\left(F(\gamma)(s)\right) + {}^{\tau} \mathcal{L}_t^{\scriptscriptstyle B}\left(\gamma(a)\right). \tag{5.32}$$

In particular, for x, x′ ^∈ ^X, we have that

$$\frac{1}{5} \bar{z}{c{t}}^{M}(x, x') \le \bar{c}{t, \tau}^{B}(x, x') \le 3\bar{c}{t, \tau}^{M}(x, x') + \bar{\mathcal{L}}_{t}^{B}(i^{B}(x)). \tag{5.33}$$

Furthermore, for every ^x ^∈ ^X, we have that

$${}^{\tau}!\mathcal{L}{t}^{B}(i^{B}(x)) = \frac{2\tau}{3}\mathcal{L}{t}^{M}(i^{M}(x)). \tag{5.34}$$

Proof. For the first inequality, let γ ∈ Γ B (x, x′ ) be such that γ : [a, b]^Z → P^B and choose a sequence of quasi-optimal points (ys) b ^s=a+1 ^∈ ^X such that

$$E_t(y_s) + \frac{1}{\tau} d(\gamma_0(s), ys)^2 \le E{t, \frac{\tau}{2}}^M(\gamma_0(s)) + \frac{\varepsilon}{b - a + 1}.$$

We then have

$$\begin{split} \sum{s=a+1}^{b-1} \frac{\tau}{2} & \mathcal{L}{t}^{M} \left( F(\gamma)(s) \right) - \varepsilon \ & \leq \sum{s=a+1}^{b-1} \left( E{t}(\gamma{0}(s)) - \underbrace{\left( E{t}(y{s}) + \frac{1}{\tau} d(\gamma{0}(s), y{s})^{2} \right)}{\geq E{t,\tau}^{B}(\gamma{0}(s), \gamma{-1}(s))} + \frac{1}{\tau} d\left( \gamma{0}(s), \gamma{1}(s) \right)^{2} \right) \ & \leq \sum{s=a+1}^{b-1} \left( E{t}(\gamma{0}(s)) - E{t,\tau}^{B} \left( \gamma{0}(s), \gamma{-1}(s) \right) \right) \ & + 4 \sum{s=a+1}^{b-1} \underbrace{\frac{1}{4\tau} d\left( \gamma{-1}(s+1), \gamma{0}(s+1) \right)^{2}}{\leq E{t} \left( \gamma{0}(s) \right) - E{t,\tau}^{B} \left( \gamma{0}(s), \gamma{-1}(s) \right) \text{ by Lemma 5.15}} \ & \leq 5 \sum{s=a+1}^{b} \left( E{t} \left( \gamma{0}(s) \right) - E{t,\tau}^{B} \left( \gamma{0}(s), \gamma{-1}(s) \right) \right) \leq 5 \sum{s=a+1}^{b} \mathcal{T}{L}^{B} \left( \gamma_{0}(s) \right), \end{split}$$

where we used Equation (5.25) in the last inequality. Since ε was arbitrary, the left inequality in (5.32) follows.

To prove the right inequality in (5.32), we likewise fix γ : [a, b]^Z → P^B , γ ∈ Γ B (x, x′ ) and choose a sequence of quasi-optimal points (ys) b ^s=a+1 such that

$$E{t,\tau}^{B}(\gamma{0}(s),\gamma{-1}(s)) + \frac{\varepsilon}{b-a+1} \ge E{t}(y{s}) + \frac{1}{\tau}d(\gamma{0}(s),y{s})^{2} - \frac{1}{4\tau}d(\gamma{-1}(s),y_{s})^{2}.$$

Owing to the estimate in Equation (5.27), we find

$$\begin{split} \sum{s=a+1}^{b-1} \bar{\mathcal{L}}^B(\gamma) - \varepsilon &\leq \sum{s=a+1}^{b-1} \left( E_t \big( \gamma_0(s) \big) - E_t(y_s) - \frac{1}{\tau} d \big( \gamma_0(s), ys \big)^2 + \frac{1}{4\tau} d \big( \gamma{-1}(s), y_s \big)^2 \ &\quad + \frac{1}{2\tau} \big( \overline{d}_s^2 + \underline{d}_s^2 \big) - \frac{1}{4\tau} \overline{d}s^2 \right) \ &\leq \sum{s=a+1}^{b-1} \left( E_t \big( \gamma_0(s) \big) - \underbrace{\left( E_t(y_s) + \frac{1}{2\tau} d (\gamma_0(s), ys)^2 \right)}{\geq E_{t,\tau}^M(\gamma_0(s))} + \frac{1}{2\tau} \underline{d}_s^2 + \frac{1}{2\tau} (\overline{d}_s^2 + \underline{d}s^2) \right) \ &\leq \sum{s=a}^{b-1} \left( Et \big( \gamma{-1}(s) \big) - E_{t,\tau}^M \big( \gamma0(s) \big) + 3 \frac{1}{2\tau} \overline{d}^2 \right) \leq 3 \sum{s=a}^{b-1} \mathcal{L}_t^M \left( F(\gamma)(s) \right), \end{split}$$

where we set d^s := d γ−1(s), γ0(s) , d^s := d γ0(s), γ1(s) and d^s := d γ−1(s), γ1(s) , and where we used Equation (5.27) in the second line. Since ε was arbitrary, the second inequality in (5.32) follows. Equation (5.33) follows directly by applying the infimum to the preceding inequality, while Equation (5.34) follows from a simple substitution into the definitions of ^τ^L ^t and ^τ^L t in Equations (4.22) and (4.29), respectively. □

The preceding lemma allows us to adapt some results from the minimizing movement scheme directly.

Lemma 5.18. Let us assume Assumptions 1 to 4. For all K ⊆ P^B compact and C > 0, there exist K′ ⊆ P^B compact such that the following implication holds for all x, x′ ∈ K and γ ∈ Γ(x, x′ ):

$$\sum{s=a+1}^{b} \mathcal{I}{t}^{B}(\gamma(s)) \leq C \implies \operatorname{Im}(\gamma) \subseteq K'. \tag{5.35}$$

In particular, for each K ⊆ P^B there exists a compact set K′ ⊆ P^B such that for every ε ∈ (0, 1) the following implication holds for all x, x′ ∈ K and γ ∈ Γ(x, x′ ):

$$\sum{s=a+1}^{b} {}^{\tau}!\mathcal{L}{t}^{B}(\gamma(s)) \leq {}^{\tau}!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!$$

Proof. For the implication in (5.35), we use the first inequality in (5.32) and the corresponding implication (5.16) in Lemma 5.10 for τ 2L M t . For (5.36), we use the second inequality in (5.33) and the continuity of ^τ c M t , τL M ^t and ^τ^L B t to find an upper bound C ′ on τ c B t (x, x′ ) for x, x′ ∈ K, and we apply (5.35) with C = C ′ + ε. □

Lemma 5.19. Let us assume Assumptions 1 to 4. Then, ^τ c B t is an action according to Definition 2.

Proof. The proof is analogous to the one for the corresponding result in the minimizing movement scheme, namely Lemma 5.11. □

We are now ready to prove that the action ^τ c B t is generated by curves.

Lemma 5.20. Let us assume Assumptions 1 to 5 and 7. Then, for all τ ≤ 1 L , the action τ c B t is generated by curves as in Definition 3.

Proof. First note that by Lemma 5.15, ^τ^L B t is nonnegative. For the continuity of E B t,τ —and thus of ^τ^L ^t — we argue as in Remark 18 and note that only points which are either in the set {^y ^∈ ^X ^| ^d(y, x) ^≤ ^d(x, x′ )} or in the set {^y ^∈ ^X ^| ^Et(y) ^≤ ^Et(x)} contribute to the infimum of

$$y \mapsto E_t(y) + \frac{1}{\tau}d(x,y)^2 - \frac{1}{4\tau}d(x',y)^2 \ge E_t(y) + \frac{1}{\tau}d(x,y)^2 - \frac{1}{4\tau}\left(d(x,y) + d(x,x')\right)^2 \tag{5.37}$$

in Equation (4.25): If y is in neither set, then the value of (5.37) is large than when using y = x as a competitor. Due to the coercivity of E^t , both those sets are compact. Thus, ^τ^L t is itself continous on P^B ×[0, T], as it is locally a supremum of equicontinous functions. As for the properties, Property 1 is immediate and Property 4 follows directly from Lemma 5.18. We now prove the remaining three properties.

Property 2: We use the first statement in Lemma 5.15 to see that the assumption

$$^{\tau}\mathcal{L}_{t}^{B}\left(\underline{x}\right)=0$$

implies that Et(x0) − E B t,τ (x0, x−1) = 0 and d(x−1, x0) = d(x0, x1). Using the second statement in Lemma 5.15, we see that he former implies that x−¹ = x0, which by the latter implies that also x−¹ = x⁰ = x¹ and thus that x = i B (x) for some ^x ^∈ ^X. By Equation (5.34), we then also have that 2τ 3L M t (i ^M (x)) = 0, which by the assumptions τ ≤ 1 L and Lemma 5.13 implies that |∂E^t |(x) = 0.

Property 3: Note that under the given assumptions, both ^τ c ^t and 2c ^t are generated by curves as in Definition 3. Furthermore, in virtue of Lemma 5.13 and Equation (5.34), under the given assumptions τ ≤ L , we have that

$$|\partial E_t|(x) = 0 \iff {}^{\frac{\tau}{2}}!\mathcal{L}_t^{\scriptscriptstyle M}(i^{\scriptscriptstyle M}(x)) = 0 \iff {}^{\tau}!\mathcal{L}_t^{\scriptscriptstyle B}(i^{\scriptscriptstyle B}(x)) = 0 \iff {}^{\tau}!\mathcal{L}_t^{\scriptscriptstyle M}(i^{\scriptscriptstyle M}(x)) = 0.$$

Using these equivalences, the "if" direction of Property 3 follows directly from the upper bound by ^τ c M t in (5.33) and the respective Property 3 of ^τ c M t . The "only if" direction follows from the lower bound by τ 2c M t in (5.33) and the respective Property 3 of τ 2c M t .

Property 5: Let K ⊆ P^B be compact and ε > 0 be chosen. By continuity of ^τ^L t , we can choose L ′ such that for all x, x′ ∈ K, the following implication holds:

$$\mathcal{T}{\mathcal{L}{t}^{B}}(\underline{x}) \leq \frac{\varepsilon}{8}, \quad d{\mathscr{P}^{B}}(\underline{x},\underline{x}') \leq L' \implies \mathcal{T}{t}^{B}(\underline{x}') \leq \frac{\varepsilon}{4}.$$ (5.38)

Since ^τ^L t is continuous and vanishes only on the diagonal of P^B , we can further pick some ε ′ ≤ 8 such that for all x ∈ K,

$${}^{\tau}!\mathcal{L}t^B\left(\underline{x}\right) \le \varepsilon' \implies \max\left{d(x{-1}, x_0), d(x_0, x_1), d(x1, x{-1})\right} \le \frac{L'}{2}. \tag{5.39}$$

Then, for any x, x′ ^∈ ^K such that ^τ^L B t (x i ) ≤ ε ′ for i ∈ {1, 2} and d(x, x′ ) ≤ L′ 2 , we set ^x^e ⁼ (x0, x1, x′ −1 ) and ^x^e ′ = (x1, x′ −1 , x′ 0 ). Then, by Equation (5.39) and using the triangular inequality, ^dP^B (xe, x) ^≤ ^L ′ and ^dP^B (x^e ′ , x′ ) ≤ L ′ . Thus, using Equation (5.38), we have that ^τ^L B t (xe) ^≤ ε 4 and ^τ^L B t (xe ′ ) ≤ ε 4 . We finish the proof by noting that (x, ^xe, ^x^e ′ , x′ ) ∈ Γ(x, x′ ). □ To finish this section, we proof Proposition 4.6.

Proof of Proposition 4.6. Let τ > 0. The fact that ^τ c B t is an action as in Definition 2 is the content of Lemma 5.19; the fact that for |∂E^t | L-Lipschitz (L > 0) and τ ≤ 1 L , τ c B t is generated by curves as in Definition 3 is the content of the above Lemma 5.20. What is left to prove is that ^τ ω¯ B ^t as defined in Definition 6 is compatible with ^τ c B ^t as in Definition 2, i.e., that for all x, x′ ^∈ ^X, (u^j )j∈^N ^∈ ^X and increasing sequences (s^j )j∈^N ^∈ ^N such that u⁰ = x, limj→∞ us^j = x ′ and

$$u_{j+1} \in \underset{y \in \mathbb{X}}{\operatorname{arg min}} \left{ E_t(y) + \frac{1}{\tau} d(y, uj)^2 - \frac{1}{4\tau} d(y, u{j-1})^2 \right},$$ (5.40)

for all j ∈ [0, ∞)Z—where we set u−¹ = x—, we have that

$$E_t(x) - E_t(x') = {}^{\tau}C_t^B(x, x').$$

However, this follows directly by applying recursively applying Equation (5.28) from Lemma 5.16 to the sequences

$$\gamma^{(i)} \coloneqq \left(i(x), (x, x, u_1), (x, u_1, u_2), (u_1, u_2, u3), \dots, (u{sj-2}, u{sj-1}, u{sj}), (u{sj-1}, u{sj}, u{sj}), i(u{s_j})\right)$$

to see that

$$\mathcal{T}{c{t}}^{B}(x, u{s{j}}) \leq E{t}(x) - E{t}(u{s{j}}) + \mathcal{T}{c{t}}^{B}((u{s{j}-1}, u{s{j}}, u{s{j}})) + \mathcal{T}{c{t}}^{B}(i(u{s{j}}))$$

We finish the proof by using continuity of E^t and ^τ c ^t and noting that the convergence of ^Et(u^j ) implies that ^τ^L t (usj−1, us^j , us^j ) + τL t i(us^j ) ^→ 0 as ^j → ∞. □

6. Numerical experiments

To illustrate our framework, we present an experiment where we simulate the breaking of an elastic rod (Figure 4)2 . The rod is modeled as a chain of n particles connected by n − 1 springs; each of the springs can either be intact or broken. The total energy of the system is a trade-off of two energies: On one hand, the potential energy V of a spring of length l is given as V (l) = ¹ 2 ^k(l−¯l) 2 if the spring is intact, and zero otherwise—where ¯l is the rest length of the spring. The surface energy S of the ith spring, on the other hand, is modelled as S = σⁱ if the spring is broken, and zero otherwise—here, σⁱ ∼ N (¯σ, ε) is the surface constant of the ith spring, where we add slight noise to break the symmetry. The total energy of the system is then given by

$$E(x) = \sum_{i \in \mathbb{B}} \sigmai + \sum{i \in \mathbb{I}} V(|xi - x{i+1}|), \tag{6.1}$$

where ^I ^⊆ [ⁿ ⁻ 1] is the set of intact springs, ^B = [ⁿ ⁻ 1] \ ^I is the set of broken springs, and ^x ^∈ ^R n is the vector of the positions of the particles.

²Videos of the simulations, as well as the full code are available at https://github.com/duesenfranz/ quasistatic\_evolutions\_simulations.

Figure 4: The simulation for an elastic rod, once for (a) δ = 1 ¹⁵ and once for (b) δ = ²⁴⁰ . From left to right, we plot (i) the initial configuration, (ii) the configuration right before the system transition which leads to the breakage of the rod, (iii) the configuration right after the transition, and (iv) the configuration at the end of the time horizon. In (a.ii), the stress is concentrated on the two end segments, which leads to the rod breaking early. This phenomenon is the result of the approximation error due to the large step size δ = 15 .

To model the transition between the intact and broken states, we introduce a latent variable ^z ^∈ [0, 1]n−¹ for each spring i. The variable zⁱ is a measure of how broken the ith spring is: zⁱ = 0 means that the spring is intact, while zⁱ = 1 means that the spring is completely broken. With this notation, the total energy of the system becomes

$$E((x,z)) = \sum_{i=1}^{n-1} z_i \cdot \sigma_i + (1-z_i) \cdot V(|xi - x{i+1}|).$$ (6.2)

While ^z ^∈ [0, 1]n−¹ theoretically allows for unphysical states where zⁱ ̸∈ {0, 1}, such states will not appear in practice: In the generic case, we expect that V (|xⁱ − xi+1|) ̸= σⁱ , in which case a critical point must have zⁱ ∈ {0, 1}.

In the simulation, we fix the first and the last particle to be at position x1(t) := (0, −t · h) and xn(t) := (1, t · h) respectively, at time t. We set the state space to be E := R ⁿ−2×2×[0, 1][n−1], since each system state is given by the positions of the inner particles x ′ := (x2, . . . , xn−1) ^∈ ^R ⁿ−2×² as well as the latent variables ^z := ^z1, . . . , zn−¹ ^∈ [0, 1]n−¹ .

In this setup, the total energy is given by

Tension in first segment $$E_t((\bar{x},\bar{z})) = \underbrace{(1-z_1) \cdot V(|x2-(0,-t\cdot h)|)}{\text{Tension in last segment}} + \underbrace{\sum_{i=2}^{n-2} (1-zi) \cdot V(|x{i+1}-xi|)}{\text{Tension in last segment}} + \underbrace{(1-z{n-1}) \cdot V(|x{n-1}-(1,t\cdot h)|)}{\text{Surface energy}} + \underbrace{\sum{i=1}^{n-1} z_i \cdot \sigmai}{\text{Surface energy}}.$$

$$(6.3)$$

Finally, we set the starting state to be the local minimum where the whole rod is intact and the points x are equidistantly spaced along the line from x1,⁰ to xn,0. The discrete quasistatic evolution follows this local minimum over time, positioning the points x equidistantly between the endpoints x1(t) and xn(t). Eventually, the potential energy of one of the springs exceeds the surface energy. At this point, this local minimum disappears and the discrete quasistatic evolution jumps to a lower energy level—the one where the spring is broken. Interestingly, this new local minimum of the rod being broken is not an isolated critical point, but a whole manifold.

Numerically, we use a BDF2 approximation of the gradient flow, corresponding to the action explained in Section 5.3, with τ = 0.1; we stop the gradient descent once the energy difference between two consecutive steps is less than 10−⁵ .

We can observe the expected convergences in the quantities involved in the energy balance: First, µ δ converges to zero everywhere but at a single point—the point where the rod breaks—as ^δ goes to zero (Figure 6a). Furthermore, ^D^δ converges to a piecewise smooth function, with a single jump point where the rod breaks (Figure 6b). The same convergences can be seen when comparing Et(η δ (t)) and R ^t ⁰ ^D^δ (s) ds for different values of δ (Figure 5).

Figure 5: Comparing Et(η δ (t)) and R ^t ⁰ ^D^δ (s) ds, for δ = 1 ¹⁵ on the left and δ = ²⁴⁰ on the right.

Figure 6: The quantities involved in the energy balanced for the elastic rod simulation, for discrete quasistatic evolutions with different values of δ. For δ going to zero, the support of µ δ collapses to the single point where the rod breaks at ^t = 0.78. Furthermore, ^D^δ converges to a piecewise smooth function, with a single jump point at this breakage time. After the rod is broken, the potential energy is constant zero for steady states and D^δ converges to zero as

Appendix A. Structure of the measure in the energy balance

δ goes to zero.

In this part, we seek to establish that the non-negative measure ¯µ appearing in the energy balance in Theorem 1 (complete) is purely atomic. Under suitable extra assumptions, this fact is expected, as it has already been observed for quasistatic evolutions in finite-dimensional Hilbert spaces constructed with limiting arguments (see, e.g., [2, 25, 21]). In this section, we adapt to our setting the arguments developed in [25, Section 5.1].

We observe that, by virtue of Assumption 8, we can rewrite the energy balance for the quasistatic evolution ˆη : [0, T] → X as follows:

$$\hat{E}(\hat{\eta}^{+}(t)) - \hat{E}(\hat{\eta}^{-}(s)) = \int{s}^{t} \partial{t} \hat{E}(\hat{\eta}(\tau)) d\tau - \bar{\mu}([s, t]). \tag{A.1}$$

Moreover, we introduce the non-increasing function ^f : [0, T] ^→ ^R as

$$f(t) := \hat{E}(\hat{\eta}(t)) - \int_0^t \partial_t \hat{E}(\hat{\eta}(\tau)) d\tau, \tag{A.2}$$

and we use the notation f+(t) := lims→t⁺ f(s). We observe right away that

$$f{+}(t) = \lim{s \to t^{+}} \left( \hat{E}(\hat{\eta}(s)) - \int{0}^{s} \partial{t} \hat{E}(\hat{\eta}(\tau)) d\tau \right) = \hat{E}(\hat{\eta}^{+}(t)) - \int{0}^{t} \partial{t} \hat{E}(\hat{\eta}(\tau)) d\tau.$$

Proposition A.1. Let us assume Assumptions 1 to 8. Let ^f : [0, T] ^→ ^R be defined as in Equation (A.2). Then, for every t ∈ [0, T), we have that

$$D{+}f{+}(t) := \liminf{h \to 0^{+}} \frac{f{+}(t+h) - f_{+}(t)}{h} \ge 0,$$

i.e., the Dini lower right derivative of f⁺ is non-negative.

Proof. For every ^s ^∈ [0, T], let us take ^x^s ^∈ ^X such that (s, xs) ^∈ ^η^ˆ ⁺(s). Then, we fix t ∈ [0, T) and we observe that

$$f{+}(t+h) - f{+}(t) = \hat{E}(\eta^{+}(t+h)) - \hat{E}(\eta^{+}(t)) - \int{t}^{t+h} \partial{t}\hat{E}(\hat{\eta}(\tau)) d\tau$$

$$= E{t+h}(x{t+h}) - E{t}(x{t}) - \int{t}^{t+h} \partial{t}\hat{E}(\hat{\eta}(\tau)) d\tau$$

$$= E{t}(x{t+h}) - E{t}(x{t}) + \int{t}^{t+h} \left(\partial{t}E{\tau}(x{t+h}) - \partial_{t}\hat{E}(\hat{\eta}(\tau))\right) d\tau$$

for every h > 0. Moreover, the expression in the intergal can be rewritten as follows:

$$\partial{t}E{\tau}(x{t+h}) - \partial{t}\hat{E}(\hat{\eta}(\tau)) = \partial{t}\hat{E}([(\tau, x{t+h})]) - \partial_{t}\hat{E}(\hat{\eta}(\tau))$$

$$= \left(\partial{t}\hat{E}([(\tau, x{t+h})]) - \partial_{t}\hat{E}(\hat{\eta}^{+}(t+h))\right)$$

$$+ \left(\partial{t}\hat{E}(\hat{\eta}^{+}(t+h)) - \partial{t}\hat{E}(\hat{\eta}^{+}(t))\right)$$

$$+ \left(\partial{t}\hat{E}(\hat{\eta}^{+}(t)) - \partial{t}\hat{E}(\hat{\eta}(\tau))\right). \tag{A.3}$$

Since [(τ, xt+h)] →^X [(t + h, xt+h)] = ˆη(t + h) as τ → t + h, it follows that the time derivative ∂tEˆ [(τ, xt+h)] ^→ ^∂tE^ˆ ηˆ ⁺(t + h) as τ → t + h, owing to Assumption 8. Moreover, recalling that τ ∈ [t, t + h], that ˆη ⁺ is right-continuous and that ˆη ⁺(^τ ) ^→^X ηˆ ⁺(t) as ^τ ^→ ^t ⁺, by virtue of Equation (A.3) and Assumption 8, we conclude that

$$\int{t}^{t+h} \left( \partial{t} E{\tau}(x{t+h}) - \partial_{t} \hat{E}(\hat{\eta}(\tau)) \right) d\tau = o(h),$$

yielding the identity

$$\liminf{h \to 0^+} \frac{f+(t+h) - f+(t)}{h} = \liminf{h \to 0^+} \frac{Et(x{t+h}) - E_t(x_t)}{h}.$$ (A.4)

Before proceeding, we need to carefully choose the representative (t, y) of ˆη ⁺(t), and to replace (t, xt) if necessary. Indeed, let us firs consider hⁿ ↘ 0 as n → ∞ such that

$$\liminf_{h \to 0^+} \frac{Et(x{t+h}) - E_t(xt)}{h} = \lim{n \to \infty} \frac{Et(x{t+h_n}) - E_t(x_t)}{h_n},\tag{A.5}$$

and such that there exits x ′ ^∈ ^X for which ^d(xt+hⁿ , x′ ) → 0 as n → ∞. We recall that it is possible to find such a sequence by virtue of Remark 15. Moreover, since [(t + hn, xt+hⁿ )] = ˆη ⁺(^t ⁺ ^hn) ^→^X ^η^ˆ ⁺(t) as ⁿ → ∞, we deduce that (t, x′ ) ∈ ηˆ ⁺(t). Hence, since [(t, x′ )] = ˆη ⁺(t) = [(t, xt)], from Equations (A.4) and (A.5) we obtain that

$$\liminf{h \to 0^+} \frac{f+(t+h) - f+(t)}{h} = \lim{n \to \infty} \frac{Et(x{t+h_n}) - E_t(x')}{h_n},\tag{A.6}$$

where we used the fact that Et(xt) = Et(x ′ ), ensured by Assumption 5. Then, we note that if Et(xt+hⁿ ) = Et(x ′ ) for infinitely many ⁿ ^∈ ^N, then the thesis follows immediately. Therefore, we assume that Et(xt+hⁿ ) ̸= Et(x ′ ) for n large enough. From this fact, since d(xt+hⁿ , x′ ) → 0 as n → ∞ and since the connected components of critical points are assumed to be well-separated (see again Assumption 5), it follows that eventually |∂E^t |(xt+hⁿ ) ̸= 0. Hence, we compute

$$\frac{Et(x{t+h_n}) - E_t(x')}{h_n} = \frac{Et(x{t+h_n}) - E_t(x')}{|\partial Et|(x{t+h_n})} \frac{|\partial Et|(x{t+h_n})}{hn} \ge -\varepsilon{x'} (d(x_{t+h_n}, x')) L,$$

where we used Assumption 6, together with the fact that |∂Et+hⁿ |(xt+hⁿ ) = 0, due to [(t+hn, xt+hⁿ )] = ˆη ⁺(t+hn). Hence, by taking the limit in the last inequality, we finally deduce that

$$\liminf{h \to 0^+} \frac{f+(t+h) - f+(t)}{h} = \lim{n \to \infty} \frac{Et(x{t+h_n}) - E_t(x')}{h_n} \ge 0$$

and we conclude the proof. □

We report below a well-known sufficient condition for monotonicity of continuous functions.

Lemma A.2. Let ^g : [a, b] ^→ ^R be continuous and such that the Dini upper right derivative of g satisfies

$$D^+g'(t) := \limsup_{h \to 0^+} \frac{g(t+h) - g(t)}{h} \ge 0$$

for every t ∈ (a, b). Then, g is non-decreasing on (a, b).

Proof. See, e.g., $$[25, Lemma 5.2]$$ .

Now, we show that the non-negative measure ¯µ appearing in the energy balance in Theorem 1 (complete) is purely atomic. Since we already took care of the peculiarity of our setting in Proposition A.1, we report the the arguments in the next proof follows the lines of [25, Theorem 5.4]. We detail the proof for the sake of completeness.

Proposition A.3. Let us assume Assumptions 1 to 8. Let µ¯ be the non-negative measure appearing in the energy balance in Theorem 3.9. Then, we have that supp µ = J, where J is the jump set of the limiting trajectory ηˆ: [0, T] → X .

Proof. We consider the non-increasing function ^f : [0, T] ^→ ^R defined as in Equa tion (A.2), and, owing to Theorem 3.9 and to the enhanced energy balance in Equa tion (A.1), we observe that the distributional derivative of f satisfies ¯µ = −df, and we recall ¯µ is a positive measure. To see that supp ¯µ = J, we introduce the non-increasing function

$$f^{J}(t) := \sum{s \in [0,t]} (f{+}(s) - f_{-}(s)),$$

and we observe that it is right-continuous. Moreover, the set of dicontinuity points for f J is exactly J, and the distributional derivative satisfies d(f J ) = (df) J , where (df) ^J denotes the jump part of the measure df. Then, we observe that ¯µ = −df ≥ −d(f J ) =: µ¯ J . In addition, recalling that f J (t) − f J (t + h) ≥ 0 for every h ≥ 0, we have that

$$\liminf{h \to 0} \frac{(f+ - f^J)(t+h) - (f+ - f^J)(t)}{h} \ge \liminf{h \to 0} \frac{f+(t+h) - f+(t)}{h} \ge 0. \tag{A.7}$$

We further notice that the function f⁺ −f J is continuous. Indeed, on the one hand, both f⁺ and f ^J are right-continuous. On the other hand, we have limh→0⁺ ^f+(^t ⁻ ^h) = ^f−(t) and

$$\lim{h \to 0^+} f^J(t - h) = \sum{s \in [0, t)} (f+(s) - f-(s)) = f^J(t) - (f+(t) - f-(t))$$

for every t ∈ (0, T], which combined provide the left-continuity of f⁺ − f J . Hence, by using Equation (A.7) and Lemma A.2, we deduce that f⁺ − f J is non-decreasing. Therefore, from the inequality f+(t) − f J (t) ≥ f+(0) − f J (0), recalling that f J (t) = −µ¯ J ([0, t]) and that f+(0) − f J (0) = f−(t), we obtain that

$$\hat{E}(\hat{\eta}^+(t)) + \bar{\mu}^J([0,t]) \ge \hat{E}(\hat{\eta}^-(0)) + \int_0^t \partial_t \hat{E}(\hat{\eta}(\tau)) d\tau$$

for every t ∈ [0, T]. Recalling that ¯µ J , µ¯ are positive, from the last inequality and from the balance in Equation (A.1) we deduce that ¯µ ^J ^≥ ^µ¯. Since by construction ¯^µ ^≥ ^µ¯ J , this shows that ¯µ = ¯µ ^J and concludes the proof. □

Appendix B. Gradient flow vs. minimizing movement scheme actions

In this section, we relate the actions ^τ c ^t and^τ c M t , corresponding to the gradient descent and to the minimizing movement scheme, respectively. From here on, we fix a time t ∈ [0, T] and consequently drop the subscript t from the notation: We concern ourselves with two actions ^τ c and ^τ c ^M and an energy functional ^E on ^X. We begin by relating ^L F and ^τ^L ^M , the instantaneous costs of the gradient flow and of the minimizing movement scheme, respectively.

Lemma B.1. Let us assume Assumptions 1 to 7. Furthermore, let us assume that L < ¹ τ , where L is the Lipschitz constant of |∂E|. Then, setting ε^L := L · τ , we have that, for every ^x ^∈ ^X,

$$(1 - \varepsilon_L)^T \mathcal{L}^M \left( i(x) \right) \le \tau \mathcal{L}^F \left( i(x) \right) \le (1 + \varepsilon_L)^T \mathcal{L}^M \left( i(x) \right). \tag{B.1}$$

Proof. We start with the first inequality in (B.1). To this end, we fix ^x ^∈ ^X, pick x ′ ^∈ ^X and ε > 0, and set ^d := ^d(x, x′ ). We use Assumption 1 to choose an absolutely continuous curve ^γ : [0, d ⁺ ^ε] ^→ ^X such that ˙^γ ^≡ 1, ^γ(0) = ^x and ^γ(^d ⁺ ^ε) = ^x ′ . Since |∂E| is a strong upper gradient—see Remark 1—we have

$$E(x) - E(x') - \frac{1}{2\tau} \underline{d}^2 \le \int_0^{\underline{d}+\varepsilon} |\partial E|(\gamma(s)) \, \mathrm{d}s - \frac{1}{2\tau} \underline{d}^2 \le \int_0^{\underline{d}+\varepsilon} \left( |\partial E|(x) + Ls \right) \, \mathrm{d}s - \frac{1}{2\tau} \underline{d}^2$$

$$\le (\underline{d}+\varepsilon) |\partial E|(x) + \frac{1}{2} L(\underline{d}+\varepsilon)^2 - \frac{1}{2\tau} \underline{d}^2.$$

Taking ε → 0, we get

$$E(x) - E(x') - \frac{1}{2\tau} \underline{d}^2 \le \underline{d} |\partial E|(x) + L \frac{1}{2} \underline{d}^2 - \frac{1}{2\tau} \underline{d}^2 = \underline{d} |\partial E|(x) - \frac{1}{2\tau} (1 - \varepsilon_L) \underline{d}^2.$$ (B.2)

The right hand side above is a quadratic equation whose maximum is attained at d = τ |∂E|(x) 1 (1−εL) ; substituting this choice of d into Equation (B.2) gives

$$E(x) - E(x') - \frac{1}{2\tau}\underline{d}^2 \le \tau \frac{1}{1 - \varepsilon_L} \frac{1}{2} |\partial E|(x)^2$$

As ^τ^L M i(x) is the supremum of the left hand side of the above equation over all x ′ ^∈ ^X, we have

$${}^{\tau}!\mathcal{L}^{M}\left(i(x)\right) \leq \tau \frac{1}{1-\varepsilon{L}} \frac{1}{2} |\partial E|(x)^{2} = \tau \frac{1}{1-\varepsilon{L}} \mathcal{L}^{F}\left(i(x)\right),$$

which finishes the proof of the first inequality.

For the second inequality in (B.1), we start by letting ^γ : [0, ^∞) ^→ ^X be a curve of maximum slope such that γ(0) = x, which in particular implies that the following equation holds for all s ≥ 0:

$$E(x) - E(\gamma(s)) = \int_0^s |\partial E|(\gamma(r))|\dot{\gamma}| dr;$$ (B.3)

such a curve of maximum slope exists by Remark 1 and [7, Prop. 2.2.3, Theorem 2.3.3]. For s > 0, we set l^s := R s 0 |γ˙(σ)| dσ and we define l^∞ := R ∞ 0 |γ˙(σ)| dσ. Note that, as γ is a curve of maximum slope, in addition to Equation (B.3), the following holds for almost all s > 0:

$$|\dot{\gamma}|(s) = |\partial E|(\gamma(s)) \le |\partial E|(x) - L \int_0^s |\dot{\gamma}|(r) dr.$$

Using Gr¨onwall's lemma, we deduce from the above that |γ˙ |(s) ≥ |∂E|(x)e ⁻Ls, and thus that

$$l_{\infty} \ge \frac{1}{L} |\partial E|(x).$$ (B.4)

As a next step, we reparametrize ^γ to obtain a curve ^γe: [0, l∞] ^→ ^X such that ^γe(0) = ^x, ˙γe(s) ^≡ 1 and such that Equation (4.14) holds for ^γ^e in place of ^γ. We calculate, for s ∈ [0, l∞],

$$\begin{split} \mathcal{T}{L}^{M}\left(i(x)\right) &\geq E(x) - E\left(\widetilde{\gamma}(s)\right) - \frac{1}{2\tau}d\left(x,\widetilde{\gamma}(s)\right)^{2} = \int{0}^{s}|\partial E|\left(\widetilde{\gamma}(r)\right)\mathrm{d}r - \frac{1}{2\tau}d\left(x,\widetilde{\gamma}(s)\right)^{2} \ &\geq \int_{0}^{s}\left(|\partial E|(x) - Lr\right)\,\mathrm{d}r - \frac{1}{2\tau}s^{2} = s|\partial E|(x) - \frac{1}{2}Ls^{2} - \frac{1}{2\tau}s^{2} \ &= s|\partial E|(x) - \frac{1}{2}\left(L + \frac{1}{\tau}\right)s^{2}. \end{split}$$

Evaluating the above at s ⋆ := |∂E|(x) L+ ¹ —justified by Equation (B.4)— gives

$${}^{\tau}\mathcal{L}^{M}\left(i(x)\right) \geq \frac{1}{2}|\partial E|(x)^{2}\tau\frac{1}{1+\varepsilon{L}} = \tau\mathcal{L}^{F}\left(i(x)\right)\frac{1}{1+\varepsilon{L}},$$

which finishes the proof of the second inequality. □

As a next step, we show that we can bound the cost of a single step of the minimizing movement scheme by the cost of the gradient descent action, as long as step is small enough.

Lemma B.2. Let us assume Assumptions 1 to 7. Furthermore, let us assume that L < ¹ τ , where L is the Lipschitz constant of |∂E|. Then, setting ε^L := L· τ , we have, for each x, x′ ^∈ ^X such that ^d(x, x′ ) ≤ τ · |∂E|(x),

$$(1 - \varepsilon_L)^{\mathsf{T}} \mathcal{L}^{M} \left( (x, x') \right) \le 2c^F(x, x') + |\partial E|(x) \left( \tau |\partial E|(x) - d(x, x') \right)$$

$$\le 2c^F(x, x') + 2\tau \mathcal{L}^F \left( i(x) \right).$$ (B.5)

In particular, if d(x, x′ ) = τ |∂E|(x), the above inequality reduces to

$$(1 - \varepsilon_L)^T \mathcal{L}^M ((x, x')) \le 2c^F (x, x').$$ (B.6)

Proof. Both the second inequality in (B.5) and (B.6) follow immediatly. In the rest of this proof, we concern ourselves the first inequality in (B.5). Under the above assumptions, we have, using Lemma B.1:

$$(1 - \varepsilon_L)^{T} \mathcal{L}^{M} ((x, x')) = (1 - \varepsilon_L) \left( \mathcal{L}^{M} (i(x)) + \frac{1}{2\tau} d(x, x')^2 \right)$$

$$\leq \tau \mathcal{L}^{F} (i(x)) + \frac{1}{2\tau} \tau^2 |\partial E|(x)^2 = 2\tau \mathcal{L}^{F} (i(x)).$$ (B.7)

On the other hand, we can bound c (x, x′ ) from below: By virtue of Lemma 5.2 it suffices to find a lower bound for R ^s 0 |∂E|(γ(σ)) dσ, uniformly over s > 0 and all absolutely continuous curves ^γ : [0, s] ^→ ^X such that ˙^γ ^≡ 1, ^γ(0) = ^x and ^γ(s) = ^x ′ . Let such s and γ be given. We note that s ≥ d(x, x′ ) =: d and write

$$\begin{split} \int_0^s |\partial E|(\gamma(\sigma)) \, \mathrm{d}\sigma &\geq \int_0^{\underline{d}} |\partial E|(\gamma(\sigma)) \, \mathrm{d}\sigma \geq \underline{d} |\partial E|(x) - \frac{1}{2} L \underline{d}^2 \ &\geq \frac{1}{2} |\partial E|(x) \underline{d} = \frac{1}{2} \tau |\partial E|(x)^2 - \frac{1}{2} |\partial E|(x) \big(\tau |\partial E|(x) - \underline{d}\big). \end{split}$$

Taking the infimum over γ in the inequality above, we obtain

$$c^{F}(x, x') \ge \tau \mathcal{L}^{F}(i(x)) - \frac{1}{2} |\partial E|(x) (\tau |\partial E|(x) - \underline{d}).$$ (B.8)

Combining Equations (B.7) and (B.8) gives

$$(1 - \varepsilon_L)^T \mathcal{L}^M ((x, x')) \le 2c^F (x, x') + |\partial E|(x) (\tau |\partial E|(x) - \underline{d})$$

as required. □

We continue by proving a inverse relation to Lemma B.2:

Lemma B.3. Let us assume Assumptions 1 to 7. Furthermore, let us assume that L < ¹ τ , where L is the Lipschitz constant of |∂E|. Then, setting ε^L := L· τ , we have, for every x, x′ ^∈ ^X:

$$c^{F}(x, x') \leq 4 \frac{1 + \varepsilon{L}}{1 - \varepsilon{L}} \mathcal{I}_{L}^{M} ((x, x'))$$

Proof. The proof is similar to the first part of the proof of Lemma B.1. We show the inequality by choosing, for an arbitrary ε > 0, as a competitor for the infimum in Equation (5.3) an absolutely continuous curve ^γ : [0, d+ε] ^→ ^X such that ˙^γ ^≡ 1, ^γ(0) = ^x and γ(d + ε) = x ′ ; the existence of such a curve is guaranteed by Assumption 1. With this choice, we have, setting d := d(x, x′ ):

$$c^{F}(x,x') \leq \int_{0}^{\underline{d}+\varepsilon} |\partial E| (\gamma(s)) \, \mathrm{d} s \leq (\underline{d}+\varepsilon) |\partial E|(x) + \frac{1}{2} L(\underline{d}+\varepsilon)^{2}.$$

Taking the limit ε → 0, we get

$$c^{F}(x,x') \leq \underline{d}|\partial E|(x) + \frac{1}{2}L\underline{d}^{2} = \underline{d}|\partial E|(x) + \frac{1}{2\tau}\underline{d}^{2} - \frac{1}{2\tau}(1 - \varepsilon_{L})\underline{d}^{2}.$$ (B.9)

If d ≤ 2τ 1−ε^L |∂E|(x), we use Lemma B.1 to reduce the above to

$$\begin{split} c^{\scriptscriptstyle F}(x,x') & \leq \frac{2\tau}{1-\varepsilon_L} |\partial E|^2(x) + \frac{1}{2\tau} \underline{d}^2 = \frac{4}{1-\varepsilon_L} \tau \mathcal{L}^{\scriptscriptstyle F}(i(x)) + \frac{1}{2\tau} \underline{d}^2 \ & \leq 4 \frac{1+\varepsilon_L}{1-\varepsilon_L} {}^{\scriptscriptstyle T}!\mathcal{L}^{\scriptscriptstyle M}\left(i(x)\right) + \frac{1}{2\tau} \underline{d}^2 \leq 4 \frac{1+\varepsilon_L}{1-\varepsilon_L} {}^{\scriptscriptstyle T}!\mathcal{L}^{\scriptscriptstyle M}\left((x,x')\right). \end{split}$$

On the other hand, if d > 2τ 1−ε^L |∂E|(x), Equation (B.9) also gives

$$c^{F}(x,x') \leq \underline{d}|\partial E|(x) + \frac{1}{2\tau}\underline{d}^{2} - \frac{1}{2\tau}(1-\varepsilon{L})\frac{2\tau}{1-\varepsilon{L}}|\partial E|(x)\underline{d}$$ $$= \frac{1}{2\tau}\underline{d}^{2} \leq {}^{\tau}\mathcal{L}^{M}\left((x,x')\right) \leq 4\frac{1+\varepsilon{L}}{1-\varepsilon{L}}{}^{\tau}\mathcal{L}^{M}\left((x,x')\right).$$

We are now in position to prove a relation between the actions ^τ c B and ^τ c ^M .

Lemma B.4. Let us assume Assumptions 1 to 7. Furthermore, let us assume that L < ¹ τ , where L is the Lipschitz constant of |∂E|. Then, setting ε^L := L· τ , we have, for any x ̸= x ′ ^∈ ^X

$$\frac{1 - \varepsilon_L}{4(1 + \varepsilon_L)} c^F(x, x') \le \tilde{c}^M(x, x') - \tilde{\mathcal{L}}^M(i(x)) \le \frac{2}{1 - \varepsilon_L} c^F(x, x') + \frac{2\tau R}{1 - \varepsilon_L}, \tag{B.10}$$

where R := supx′′∈K^X L i(x ′′) , K ⊆ P^F is a chosen compact such that Property 4 for c is fulfilled for K, and K^X denotes the projection of K onto X.

Proof. We start with the first inequality. By taking the infimum, it suffices to show that for all competitors γ ∈ Γ M i(x), i(x ′ ) , we have that

$$\frac{1-\varepsilon_L}{4(1+\varepsilonL)}c^F(x,x') \le \sum{j=a+1}^{b} {}^{\tau}!\mathcal{L}^{\scriptscriptstyle M}\left(\gamma(j)\right) - {}^{\tau}!\mathcal{L}^{\scriptscriptstyle M}\left(i(x)\right) = \sum_{j=a+1}^{b-1} {}^{\tau}!\mathcal{L}^{\scriptscriptstyle M}\left(\gamma(j)\right), \tag{B.11}$$

where a < b ^∈ ^Z are chosen such that Dom(γ) = [a, b]Z; for the second equality above, note that necessarily γ(b) = i(x ′ ). However, (B.11) follows by repeatedly applying Lemma B.3, using the triangular inequality on c F and noting that γ2(j) = γ1(j + 1) for all j ∈ [a, b − 1]^Z and that γ1(a + 1) = x, γ2(b − 1) = x ′ .

We now turn to the second inequality in (B.10), which is a little more intricate to prove. We start by picking some ε > 0 and a curve γ ∈ Γ F i(x), i(x ′ ) such that R b a L F γ(s) ds ≤ c F (x, x′ ) + ^ε, where a < b ^∈ ^R are such that Dom(γ) = [a, b]R, and such that Im(γ) ⊆ K—Property 4 guarantees that we can indeed pick such a curve. In the rest of this proof, we will choose appriate control points along the curve γ to repeatedly apply Lemma B.2 to the segments of γ.

Recalling the indexing conventions on γ introduced in Proposition 4.4, we pick some arbitrary ε ′ > 0 and iteratively choose controlpoints a = t¹ < · · · < tⁿ ∈ [a, b] such that—setting yⁱ := γx(ti)—d(yⁱ , yi+1) = max{τ |∂E|(yi), ε′} for 1 ≤ i < n and such that d(yn, x′ ) ≤ τ |∂E|(yn). Denoting the length of γ^x by l, we have that n ≤ ε ′ . For each

1 ≤ i < n, if τ |∂E|(yi) ≤ ε ′ , we can use Lemma B.1 to see that

$$(1 - \varepsilon_L)^{\tau} \mathcal{L}^{M} \left( (yi, y{i+1}) \right) \leq \tau \mathcal{L}^{F} \left( i(y_i) \right) + \frac{1 - \varepsilon_L}{2\tau} \varepsilon'^{2} = \frac{1}{2\tau} \left( \tau |\partial E|(y_i) \right)^{2} + \frac{1 - \varepsilon_L}{2\tau} \varepsilon'^{2}$$ $$\leq \frac{1}{2\tau} \varepsilon'^{2} + \frac{1}{2\tau} \varepsilon'^{2} = \frac{1}{\tau} \varepsilon'^{2}.$$

On the other hand, if τ |∂E|(yi) ≥ ε ′ , we can use Lemma B.2 to see that (1−εL) τL M (yⁱ , yi+1) ≤ 2c F (yⁱ , yi+1). Thus, either way,

$$(1 - \varepsilon_L)^{\tau} \mathcal{L}^M \left( (yi, y{i+1}) \right) \le 2c^F \left( yi, y{i+1} \right) + \frac{1}{\tau} \varepsilon'^2 \le 2 \int_{ti}^{t{i+1}} \mathcal{L}^F \left( \gamma(s) \right) \mathrm{d}s + \frac{1}{\tau} \varepsilon'^2$$

furthermore, also using Lemma B.2, we also have that

$$(1 - \varepsilon_L)^{\mathsf{T}} \mathcal{L}^{\mathsf{M}}\left((yn, x')\right) \leq 2c^{\mathsf{F}}(x, x') + 2\tau \mathcal{L}^{\mathsf{F}}\left(i(x)\right) \leq 2\int{t_n}^{b} \mathcal{L}^{\mathsf{F}}\left(\gamma(s)\right) ds + 2\tau \mathcal{L}^{\mathsf{F}}\left(i(x)\right).$$

Using the curve i(x),(y1, y2),(y2, y3), . . . ,(yn−1, yn),(yn, x′ ), i(x ′ ) ∈ Γ M i(x), i(x ′ as a competitor for the infimum in ^τ c ^M (x, x′ ), we finally get

$$\begin{split} \tilde{c}^{M}(x,x') &\leq \sum{i=1}^{n-1} \tilde{\mathcal{L}}^{M}\left(\left(y{i},y{i+1}\right)\right) + \tilde{\mathcal{L}}^{M}\left(\left(y{n},x'\right)\right) + \tilde{\mathcal{L}}^{M}\left(i(x')\right) \ &\leq \frac{1}{1-\varepsilon{L}} \left(2\sum{i=1}^{n-1} \int{t{i}}^{t{i+1}} \mathcal{L}^{F}\left(\gamma(s)\right) \, \mathrm{d}s + (n-1)\frac{1}{\tau}\varepsilon'^{2} + \ &\qquad \qquad 2\int{t{n}}^{b} \mathcal{L}^{F}\left(\gamma(s)\right) \, \mathrm{d}s + 2\tau \mathcal{L}^{F}\left(i(x)\right)\right) + \tilde{\mathcal{L}}^{M}\left(i(x')\right) \ &\leq \frac{1}{1-\varepsilon{L}} \left(2\int{a}^{b} \mathcal{L}^{F}\left(\gamma(s)\right) \, \mathrm{d}s + \frac{l\varepsilon'}{\tau} + 2\tau R\right) + \tilde{\mathcal{L}}^{M}\left(i(x')\right) \ &\leq \frac{2}{1-\varepsilon{L}}c^{F}\left(x,x'\right) + \varepsilon + \frac{l\varepsilon'}{\tau} + \frac{2\tau R}{1-\varepsilon_{L}} + \tilde{\mathcal{L}}^{M}\left(i(x')\right). \end{split}$$

As both ε > 0 and ε ′ > 0 were arbitrary, we can take the limit ε → 0 and ε ′ → 0 in the above inequality to obtain the thesis. □

Acknowledgments

The work of SA was partially funded by the Austrian Science Fund through the project 10.55776/P35359, by the University of Naples Federico II through the FRA Project "ReSinApas", by the MUR - PRIN 2022 project "Variational Analysis of Complex Systems in Materials Science, Physics and Biology", No. 2022HKBF5C, funded by European Union NextGenerationEU, and by the Gruppo Nazionale per l'Analisi Matematica, la Probabilita e le loro Applicazioni (GNAMPA-INdAM, Project 2025: DISCOV-ERIES - Difetti e Interfacce in Sistemi Continui: un'Ottica Variazionale in Elasticita con Risultati Innovativi ed Efficaci Sviluppi). AS acknowledges partial support from GNAMPA-INdAM.

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