PERTURBATIONS OF MINIMIZING MOVEMENTS AND CURVES OF MAXIMAL SLOPE

. We modify the De Giorgi’s minimizing movements scheme for a functional φ , by perturbing the dissipation term, and ﬁnd a condition on the perturbations which ensures the convergence of the scheme to an absolutely continuous perturbed minimizing movement. The perturbations produce a variation of the metric derivative of the minimizing movement. This process is formalized by the introduction of the notion of curve of maximal slope for φ with a given rate. We show that if we relax the condition on the perturbations we may have many diﬀerent meaningful eﬀects; in particular, some perturbed minimizing movements may explore diﬀerent potential wells.


Introduction. The method of minimizing movements was introduced by De
Giorgi to define a notion of evolution under very weak hypotheses. It consists in introducing a time-discretization scale and a corresponding time-discrete curve by solving an iterative Euler-type scheme. By refining the time scale we obtain a continuous curve.
Ambrosio, Gigli, and Savaré developed this method in [4] to formulate a notion of gradient flow in a complete metric space (S, d) for a given proper functional φ : S → (−∞, +∞]. They considered a time discretization τ = {τ n }, of amplitude |τ | which tends to zero, and defined a time-discrete motion U τ : [0, +∞) → S (called a discrete solution) by interpolating a sequence (U τ n ) which solves the recursive scheme U τ n ∈ argmin u∈S φ(u) + d 2 (u, U τ n−1 ) 2τ n , n ≥ 1, starting from a given initial datum U τ 0 . The minimization is localized, through the dissipation term d 2 (u, U τ n−1 )/2τ n , in a neighborhood of the previous step of amplitude depending on τ . Under suitable assumptions on φ, when |τ | → 0, the discrete solutions converge to an absolutely continuous curve U : [0, +∞) → S (a minimizing movement). Denoting with |U | the metric derivative of U , and with |∂ − φ| the relaxed metric slope of φ (Definition 2.6 and 3.1), they also proved that these minimizing movements are curves of maximal slope for φ; i.e., for every t ≥ s ≥ 0, φ(U (t)) − φ(U (s)) ≤ − 1 2 t s |U | 2 (ξ)dξ − 1 2 A few years earlier, Jordan, Kinderlehrer, and Otto had used minimizing movements in [12] to study Fokker-Plank equations in which the drift term was the gradient of a potential field; they proved that minimizing movements, obtained by using Wasserstein metric as the dissipation, were solutions of the Fokker-Plank equation. This work was generalized in [4] to build a theory of Wasserstein gradient-flows in the space of probability measures.
The De Giorgi's idea has been adopted also by Almgren, Taylor, and Wang to study motion by mean curvature of boundaries in R n in their seminal work [3], which started a vast amount of literature.
We will introduce a variation of the method described above where the amplitude of the neighborhoods of minimization depends not only on τ , but also on a given sequence of positive coefficients (a τ n ), which we call perturbation. In this paper, we will consider a uniform time-discretization for simplicity, but every regular partition of the positive half-line can be considered; with an abuse of notation, we will denote τ = |τ |. We will modify the scheme by multiplying these coefficients to the dissipation, mimicking the perturbation effect of a noise term. Hence, we will consider discrete solutions u τ : [0, +∞) → S which interpolate sequences (u τ n ) solving u τ n ∈ argmin u∈S φ(u) + a τ n d 2 (u, u τ n−1 ) 2τ , n ≥ 1, on a uniform time-partition of amplitude τ . The sequences (u τ n ) are equal to those defined in [4], previously denoted as (U τ n ), obtained considering τ n = τ /a τ n . Nevertheless, the interpolation curves u τ and U τ are made on different time-discretizations, therefore they converge to different motions. A limit of the discrete solutions of this scheme will be called perturbed minimizing movement. We will also note that, if the perturbations are regular enough, we can apply directly the classical method of Ambrosio, Gigli, and Savaré, and obtain the perturbed minimizing movements through a change of variable. However we will consider very general hypotheses on the perturbations, and this allows us to use an analogous method, slightly modifying the classical one. These perturbations may also be seen as a variation of the functional, considering φ(u)/a τ n in the minimization problem. For the interested readers we suggest the work by Fleissner and Savaré [11].
Following the results of [4], we will prove that, under suitable hypotheses on φ, if the perturbations are such that the inverses are locally uniformly integrable, discrete solutions converge to an absolutely continuous perturbed minimizing movement (Theorem 2.4). We will show that these minimizing movements satisfy the energy estimate φ(u(t)) − φ(u(s)) ≤ − 1 2 t s a * (ξ)|u | 2 (ξ)dξ − 1 2 t s 1 a * (ξ) |∂ − φ| 2 (u(ξ))dξ for every t ≥ s ≥ 0, where a * is a function such that 1/a * is a weak limit in L 1 loc of {1/a τ }. Therefore we will say that perturbed minimizing movements are curves of maximal slope for φ with rate 1/a * , provided that |∂ − φ| is a strong upper gradient (Theorem 3.9).
By means of many different examples, we will see that, if some of the conditions on the perturbations are not satisfied, discrete solutions may diverge, or converge to a discontinuous curve. We will show that this condition can be relaxed, renouncing to the continuity of the perturbed minimizing movements, which in general may be assumed to be piecewise absolutely continuous. This leads us to observe that perturbed minimizing movements for multi-well energy functionals may explore local minima, while in the classical case the motion would be confined in a single potential well (Example 4.4 and 4.6).
Recently, the method of minimizing movements expounded in [4] has been applied to a family of functionals {φ ε } instead of a single one, so that the discrete solutions {u τ,ε } depend also on ε. Conditions which ensure the convergence of the discrete solutions to a curve of maximal slope for the Γ-limit of the energies, as τ and ε tend to zero, was exhibited in particular cases, for instance by Sandier and Serfaty in [13] and Colombo and Gobbino in [8]. While a wider treatment has been given by Braides, Colombo, Gobbino, and Solci in [7], or by Fleissner in [10]. The general case may present different limits, corresponding to the relation between the two small parameters ε and τ , as shown by Braides in [6], and precised for oscillating potentials by Ansini, Braides and Zimmer in [2] (see also [1]). The perturbation approach may be applied also in these cases, but it will not be treated in this paper.
2. Perturbed minimizing movements. Following the notation in [4], let (S, d) be a complete metric space, and let σ be a Hausdorff topology, weaker than the one induced by the metric, and such that d is σ-lower semicontinuous.
Fixed a positive constant τ * , for every τ ∈ (0, τ * ), which stands for the timediscretization scale, we consider a sequence (a τ n ) ∞ n=1 of strictly positive real numbers. We call any such sequences (with τ fixed) a perturbation, and we will extend it to the function a τ : (0, +∞) → (0, +∞), a τ (t) := a τ t τ . (1) We consider a functional φ : S → (−∞, +∞], sometimes called the energy functional, and we assume that it is proper; i.e., it is not identically equal to +∞. For τ ∈ (0, τ * ), we denote as (u τ n ) ∞ n=0 any sequence solving the recursive minimum problem for a given initial datum u τ 0 . If such a sequence exists, it is called a discrete solution for the implicit Euler-type scheme along φ at time-discretization scale τ with perturbation a τ , and initial datum u τ 0 . This sequence is identified with the corresponding interpolation curve u τ : [0, +∞) → S, u τ (t) := u t τ . The n-th element u τ n of a discrete solution is called a discrete step, or simply a step. In the case that a τ is the constant function 1, the scheme (2) is equal to the recursive scheme (2.0.4) presented in [4]. Definition 2.1. If there exist a sequence {τ k } ⊂ (0, τ * ) which tends to zero such that for every k there exists a discrete solution u τ k for the scheme (2), and a curve u : [0, +∞) → S such that u τ k (t) σ-converges to u(t) as k → +∞, for all t ≥ 0, then u is called a (generalized) {a τ }-perturbed minimizing movement for φ.
We sometimes say that u is a (a τ k )-perturbed minimizing movement for φ when we want to highlight the role of the sequence (τ k ).
2.1. Basic assumptions. We consider the following hypotheses for the energy functional φ. Lower semicontinuity and compactness hypotheses are the same as those considered in Section 2.1 [4]; whereas, we have to slightly modify the coercivity assumption, in order to consider the case in which a τ n /τ are not equibounded from below: H1 (lower semicontinuity) φ is σ-lower semicontinuous; H2 (coercivity) there exists u * ∈ S such that for any constant c > 0 Remark 1. The employ of the auxiliary topology σ allows us to use scheme (2) for a wide class of functionals. Indeed, if S = X is a reflexive Banach space, every weakly lower-semicontinuous functional satisfies the compactness hypothesis for the weak topology, but not for the norm.
The previous hypotheses ensure the following result (a modification of Lemma 2.2.1 and Corollary 2.2.2 [4]).
Proof. Fixed τ and n ≥ 1, we suppose that there exist the first n − 1 steps (u τ i ) n−1 i=0 of a discrete solution. For any v ∈ S, we consider the functional Φ τ,n,v (u) = φ(u) + a τ n d 2 (u, v) 2τ whose minima are the n-th step of a discrete solution, when v = u τ n−1 . Since φ and d are σ-lower semicontinuous, then Φ τ,n,v satisfies the same property too. Let u * be as in H2, by the triangular inequality and Young's inequality, we have . Therefore, by the coercivity assumption H2, we get From the previous formula, we have that so that the sublevels {u ∈ S | Φ τ,n,v (u) ≤ c} ⊂ {u ∈ S | φ(u) ≤ c} are bounded; hence σ-precompact by hypothesis H3. The result follows by applying the Weierstrass theorem.
If we consider perturbations {a τ } as in (1) regular enough (e.g. bounded and a τ ≥ α > 0), we can directly apply the method of Ambrosio, Gigli, and Savaré to the scheme (2) to approach the problem (see Remark 4). Nevertheless, for more general perturbations, the application of the classical method is not immediate or not possible. Moreover we want to distinguish the role of the coefficients a τ n from the one of the time-discretization scale, in order to highlight the perturbation effect on the dissipation. Hence, in the following, we will recall the main results presented in [4].
Our aim is to use Proposition 3.3.1 [4], which is a generalization of the Ascoli-Arzelá theorem, to obtain the convergence of the discrete solutions; i.e., the existence of a perturbed minimizing movement. For the reader's convenience, we recall the result below. where I is a discrete set. If {v τ } are such that In order to apply Lemma 2.2 to the discrete solutions, they must satisfy assumptions (i) and (ii), which replace the usual equiboundedness and equicontinuity properties of the Ascoli-Arzelá theorem. Hence we add the following hypotheses: H4 (control of initial data) there exists a constant C 0 such that d(u τ 0 , u * ) ≤ C 0 and φ(u τ 0 ) ≤ C 0 ; H5 (local uniform integrability) the family {1/a τ } is uniformly integrable in [0, T ] for all T > 0.
Remark 2. Assumption H5 implies that the family {1/a τ } is weakly convergent, up to subsequences, in L 1 loc (0, +∞) by the Dunford-Pettis theorem. We denote as a * : (0, +∞) → [0, +∞] any measurable function such that 1/a * is a weak limit for {1/a τ }, with the assumption that, if 1/a * (t) = 0 or +∞ then a * (t) = +∞ or 0, respectively. This notation is inspired by the fact that periodic perturbations, which oscillate between two or more values, weakly converge to the function that constantly assume the value of the inverse of the harmonic mean, sometimes denoted by a * .
Note that, by the local uniform integrability, {1/a τ } is uniformly bounded in L 1 (0, T ) for every T > 0; hence we may define

2.2.
Regularity of discrete solutions. Assumptions H4 and H5 provide to the discrete solutions the regularity properties (i) and (ii) of Lemma 2.2. Before proving it, we note first that the energy functional φ has a decreasing behavior along any discrete solution (u τ n ). In fact, setting u = u τ n−1 in the n-th minimization problem of scheme (2), we have This inequality also leads us to observe that the increments of the discrete solutions have an upper bound First, we recall a useful discrete version of the Gronwall Lemma.
). Fixed an integer N, for any 1 ≤ n ≤ N , let b n , τ n ∈ [0, +∞), and let A and α be two positive constant such that ατ n < 1/2, Proposition 2 (Equicompactness of discrete orbits). Let φ satisfy assumption H2, let {u τ 0 } be initial data satisfying H4. Let {a τ } be a family of perturbations as in (1) such that {1/a τ } is locally L 1 -equibounded, and let C 0,T defined as in Remark 2. If there exists a discrete solution u τ , then for every T > 0 there exists a positive constant C T such that where C 0 is the same as in H4. In addition, if hypothesis H3 is satisfied, the set of all discrete orbits {u τ (t) | t ∈ [0, T ], τ ∈ (0, τ * )} is σ-precompact.
Proposition 3 (Equicontinuity of discrete solutions). Let φ satisfy assumption H2, and let the initial data {u τ 0 } and the constant C 0 be as in H4. Let {a τ } be a family of perturbations as in (1) If there exists a discrete solution u τ , then for every T > 0 there exist a constant C = C(C 0 , T ) and a function such that Proof. We set n = t/τ and m = s/τ (for simplicity we consider t > s). Applying the triangular inequality and the discrete Holder's inequality to (4) we have that By the coercivity condition H2, the energy is bounded from below on bounded sets, so by the first of (5) we get inf{φ(u so that, by (7) we get Denoting C := 2 √ C 0 − m T we obtain the thesis. Proof. By Proposition 1, there exist {u τ } discrete solutions for every τ ∈ (0, τ * ). We consider the restriction to [0, 1] of u τ . As a consequence of H5, θ T defined in (6) is a modulus of continuity. By Proposition 2 and 3 we can apply Lemma 2.2. Hence, there exists a sequence (τ 1,k ) such that u τ 1,k (t) σ-converges to u 1 (t) for all t ∈ [0, 1]. Note that, since θ T is a modulus of continuity, the set I defined in Lemma 2.2 is empty, therefore the limit is continuous. Inductively, we can consider Hence we can extract a subsequence u τ k := u τ k,k σ-converging to a continuous perturbed minimizing movement u : [0, +∞) → S.
Hypothesis H5 imposes an additional regularity to the perturbed minimizing movements, which actually are absolutely continuous. To obtain it, we have to recall the notion of discrete derivative for discrete piecewise-constant functions. We also recall the definition of absolutely continuous curve in complete metric spaces and of its metric derivative (see e.g. Definition 1.1.1 and Theorem 1.1.2 [4]).
Hence, for any discrete solution u τ of the scheme (2), taking t n = nτ , we denote its discrete derivative as Definition 2.6. Let (S, d) be a complete metric space, and let v : (a, b) → S be an absolutely continuous curve; i.e. there exists m ∈ L 1 (a, b) such that This limit is defined almost everywhere, and it coincides with the minimal m ∈ L 1 (a, b) satisfying the previous inequality.
Proof. Let θ T be defined as in (6), and θ + Integrating the discrete derivatives defined as in (8) in an interval (s, t), and reasoning as in proof of Proposition 3, by the uniform integrability of for every 0 ≤ s < t < T , where n = t/τ k , m = s/τ k . This yields the uniform integrability of the discrete derivatives; i.e., their weak compactness in L 1 (0, T ) which proves (i). By formula (8) we have that Taking the limit, by the σ-lower semicontinuity of d and the weakly convergence of the discrete derivatives proved at point (i), we get so that u ∈ AC loc (0, +∞; S). Therefore the metric derivative |u | exists almost everywhere, and for its minimality |u |(t) ≤ A(t) for almost every t ≥ 0.
3. Curves of maximal slope with a given rate. This section is devoted to proving that, under suitable assumptions on φ, the perturbed minimizing movements as in Definition (2.1) are curves of steepest descend for the functional φ, in a sense that will be precised in the following. This is a generalization of Theorem 2.3.3 [4], but the presence of the perturbations yields a variation of the velocity of the curves. First, we remind the crucial concept of strong upper gradient for a functional (see e.g. Definition 1.2.1 [4]).
Definition 3.2 (Curve of maximal slope with a given rate). Let φ : S → (−∞, +∞] be a proper functional, let λ : (a, b) → [0, +∞] be a measurable function, and assume that 1/λ(t) = +∞ or 0 when λ(t) = 0 or +∞ respectively. A curve u ∈ AC(a, b; D(φ)) is a curve of maximal slope for φ with respect to a strong upper gradient g with rate λ if φ•u is equal almost everywhere to a nonincreasing function ϕ in (a, b), and for all a < s ≤ t < b In order to simplify the notation, we assume that zero multiplied by +∞ is null in the integral inequality, and that φ • u = ϕ in all (a, b); hence we will always write the following compact form Note that, if λ ≡ 1, u is a curve of maximal slope for φ with respect to g, according to the classical definition given by Ambrosio, Gigli, and Savaré.
Applying Young's inequality to (9) and the definition of strong upper gradient we get ). Furthermore, in Young's inequality the equal sign holds if and only if the terms are the same, so that every curve of maximal slope with rate λ satisfies the following metric gradient-flow; for almost every t ∈ (a, b).
As mentioned before, we will prove that perturbed minimizing movements are curves of maximal slope for φ with respect to |∂ − φ| with a rate depending on the perturbations {a τ }.
3.1. The Moreau-Yosida approximation scheme. In order to obtain the energy estimate for a perturbed minimizing movement, we will prove that discrete solutions satisfy an energy estimate as well, and then taking the limit as τ → 0, obtain (9). In the following two sections, we introduce an approximation scheme, analogous to the one presented in Chapter 3 [4], to work with discrete solutions.
We also have a slope estimate; applying Lemma 3.1.3 [4] with τ = δ/a τ n we get These properties will be very useful in the following.

De Giorgi's interpolants.
To obtain the discrete energy estimate mentioned before, we use the De Giorgi's interpolation argument. Mimicking Definition 3.2.1 [4], we give the following notions.
Proof. Fixed t ∈ [0, T ] with t = (n − 1)τ + δ. Let θ T be defined as in (6). By (15) and the second of formula (11) we get for τ small enough. Moreover, fixed t, s ∈ [0, T ], by Proposition 3 and (17) we obtain Therefore we have proved that (ũ τ k ) satisfies hypotheses (i) and (ii) of Lemma 2.2. For any converging subsequence (u τ k ), let v be its σ-pointwise limit, we get and the thesis follows.
3.3. Perturbed minimizing movements are curves of maximal slope. Using De Giorgi's interpolation scheme, we have the following a priori energy estimate for the discrete solutions.
Proposition 6. Let φ satisfy assumptions H1-H3, let {a τ } be perturbations as in (1), and suppose that there exists {u τ } a family of discrete solutions. For every n ≥ 1 and τ we have Proof. Integrating (13) on the interval (δ, τ ) we get Then, taking the limit for δ 0 in (19), by the first of (12), for every i ≥ 1 and τ ∈ (0, τ * ), we have that Choosing u = u τ i−1 and v = u τ i we get Taking the sum for all i from 1 to n we get the thesis.
As in [4], the result that perturbed minimizing movements are curves of maximal slope with a given rate is obtained by taking the limit in the discrete energy estimate (18) as τ → 0. Nevertheless, the presence of the perturbation terms prevent from taking the limit directly. To work around this problem we need the two following results.
Proof. The existence of a (a τ k )-perturbed minimizing movement is provided by Theorem 2.4. By the monotonicity of φ(u τ ), the σ-lower semicontinuity of φ, and Helly's lemma we get lim k→+∞ φ(u τ k ) ≥ φ(u(t)). Let (τ k ) be a subsequence of (τ k ) such that Proposition 4, Lemma 3.7 and 3.8 hold. Then by Lemma 3.7 and 3.8, we have that By Definition 3.2 we get the thesis.
Remark 4. The case in which the perturbations {a τ } defined in (1) have inverses that are globally uniformly integrable, and ∞ 0 1/a τ (t)dt = +∞, can be studied directly applying the method of Ambrosio, Gigli, and Savaré.
In [4] a sequence of positive coefficients (τ n ), of amplitude |τ | := sup n τ n < +∞, is used as a time-discretization scale, provided that n τ n = +∞. A sequence (U τ n ) which solves is called a discrete solution, starting from an initial datum U τ 0 ∈ D(φ). We will refer to this as a classical discrete solution, to distinguish it from the perturbed one. If we consider τ n = τ /a τ n , the assumptions on (τ n ) are satisfied. By the change of parameter ϕ τ (t) = t 0 1/a τ (ξ)dξ we can pass from a classical discrete solution to a discrete solution u τ of the scheme (2) defined as u τ (t) = U τ (ϕ τ (t)), for every t ≥ 0. In [4] it is proved that, if assumptions H1-H4 hold, discrete solutions pointwise σconverge (up to subsequences) to a classical minimizing movement U as |τ | → 0. Moreover, it is absolutely continuous and satisfies the energy estimate for every s ≥ 0, provided that the relaxed slope is a strong upper gradient; hence U is a curve of maximal slope for φ with respect to |∂ − φ|.
By the uniform integrability of the perturbations, the family {ϕ τ } is equicontinuous, so (up to subsequences) it uniformly converges to a limit ϕ(t) = t 0 1/a * (ξ)dξ. This proves the existence of an absolutely continuous perturbed minimizing movement u = U • ϕ, and its metric derivative satisfies |u |(t) = |U |(ϕ(t))/a * (t). Now, changing variable in the energy estimate with ξ = ϕ(ζ), we have that |∂ − φ| 2 (u(ζ))dζ and obtain the result of Theorem 3.9; i.e., perturbed minimizing movements are curves of maximal slope for φ with respect to |∂ − φ| with rate 1/a * . Now, we present three examples of perturbed minimizing movements in order to show the effects of the perturbations in well known frameworks.
Example 3.10. We consider S = R. Let φ(t) = t 2 be the energy functional, and let {u τ 0 } be a family of initial data converging to u 0 as τ → 0. We consider perturbations oscillating between two positive parameters 0 < α ≤ β a τ n = α n odd β n even.
The family {1/a τ } weakly* converges in L ∞ (0, +∞) to its average; i.e, 1/a * = (α −1 + β −1 )/2, that is the inverse of the harmonic mean between α and β. All the hypotheses of Theorem 3.9 are satisfied, therefore there exists a gradient flow. It is the solution of u = −2u/a * starting from u 0 , that is u(t) = u 0 e −2t/a * . Discrete solutions u τ are pictured in Figure 1. Note that we may consider divergent coefficients a τ n . They may produce a constant motion as in the following example.
Let {a τ } be any family of perturbations satisfying H5, and let 1/a * be a weak limit. Recalling that, in Banach spaces, the metric derivative of an absolutely continuous curve is the norm of its derivative, then by Theorem 3.9 we get the perturbed gradient flow for almost every t ≥ 0, starting from u 0 ∈ H 1 0 (Ω), which solves for almost every t > 0 in the distributional sense. In this case, the perturbation term 1/a * takes the place of the thermal diffusivity coefficient in the classical heat equation. Nevertheless while the thermal diffusivity is a constant, 1/a * changes in time.

4.
Relaxing the condition on the perturbations. Hypothesis H5 plays a crucial role for the equicontinuity of the discrete solutions, and hence for their convergence to an absolutely continuous perturbed minimizing movement. Considering perturbations that do not satisfy it could create a lack of convergence or continuity, as it is shown by the next two examples in R.
Example 4.1 (Lack of convergence). We consider the functional φ(t) = −t, and any bounded family of initial values. For the sake of simplicity we consider u τ 0 ≡ 0, so that assumptions H1-H4 hold.
Example 4.2 (Lack of continuity). We consider the functional φ(t) = t 2 /2, and initial data u τ 0 converging to u 0 = 0, otherwise we have a trivial motion because 0 is the global minimum of the energy. We consider the following perturbations For such perturbations, assumption H5 is not satisfied. In fact, taking E τ = I τ ∩ (0, 1], whose Lebesgue measures go to zero, we have that Eτ 1/a τ (t)dt = 1 for every τ , so that the uniform integrability is not satisfied. In this case, the n-th step of discrete solutions of the scheme (2) is equal to 1+τ u τ n−1 otherwise, and the discrete solutions are Even if H5 does not hold, we still have the convergence of (u τ ). In this case we lose the continuity of the limit motion. In fact, taking the limit as τ → 0, we obtain the perturbed minimizing movement u(t) = u 0 2 − t e −t , which is a piecewise absolutely continuous curve.
Remark 5. Note that Theorem 3.9 can be applied even if some coefficients a τ n tend to zero. As an example, slightly modifying the previous perturbations as a τ (t) = τ α t ∈ I τ 1 otherwise, with α ∈ (0, 1), assumption H5 is satisfied, because E 1/a τ (t)dt ≤ |E|τ 1−α for every measurable set E. Hence, (1/a τ ) is locally uniformly integrable, and weakly converges to 1. Therefore, we can apply Theorem 3.9. Indeed Figure 3. The graphs represent two discrete solutions for the same value of τ , corresponding respectively to perturbations as in (25) and (26). Note the discontinuous behavior on the left, while on the right jumps are going to disappear.
Taking the limit as τ → 0 we get u(t) = u 0 e −t .
The previous example suggests that, renouncing to the continuity of the perturbed minimizing movements, hypothesis H5 could be replaced by a relaxed one, which however ensures the convergence of the discrete solutions. Now, we consider the following assumption on {a τ }; H5 ' there exists a set of isolated points I = {t j } ⊂ [0, +∞), and a family {I τ } of sets unions of intervals with endpoints in τ Z, pointwise converging to I as τ → 0, such that {1/a τ χ [0,+∞)\Iτ } is locally uniformly integrable in [0, +∞), and {1/a τ } is locally equibounded in the L 1 -norm.
By substituting H5 with H5 ' , we are considering more general perturbations, which can violate the local uniform integrability in some intervals of the time discretization that accumulate around some isolated points, as for instance (a τ n ) defined in (25). Note that we cannot renounce to the local L 1 -equiboundedness which is crucial in the proof of Proposition 2, as Example 4.1 shows. Moreover, let u be a (a τ k )-perturbed minimizing movement, if |∂ − φ| is a strong upper gradient and the following compatibility conditions hold , for every t j ∈ I, then u is a curve of maximal slope for φ with respect to |∂ − φ| with rate 1/a * , in (t j , t j+1 ), starting from u(t + j ), for every t j ∈ I, and in (0, t 1 ) starting from u 0 .
Proof. Fixed T > 0, by the L 1 -equiboundedness of {1/a τ }, we can apply Proposition 2 and 3, for all t ∈ [0, T ]. By H5 ' , θ T defined in (6) hence the jumps are finite, and the result follows. Then let us define v τ j (t) := u τ (t − t j ) for every t j ∈ I. For any v τ j we can apply Theorem 3.9 in (0, t j+1 − t j ), and we get the thesis.
Remark 6. In Theorem 4.3, it is not specified what happens in the points t j ∈ I. This because the convergence of u τ (t j ) depends on the convergence of I τ to I. More precisely, in Example 4.2 the {a τ }-perturbed minimizing movement corresponding to the perturbations defined as in (25) is continuous from the right; i.e., u τ (t j ) → u(t + j ). Nevertheless, if we consider {a τ } corresponding to I τ + τ , which still converges to I, the discrete solutions would be , and u is continuous from the left. In the following, we will not be interested in the convergence on the points of I, so for the sake of simplicity we will deal with a specific kind of perturbations satisfying assumption H5 ' , as follows.
We can generalize these perturbations, considering a bounded sequence of positive coefficients (δ j ) such that a τ (t j ) = δ j τ , for every t j ∈ I, and perturbations {b τ } satisfying H5 such that a τ (t) = b τ (t), for every t ∈ I τ .
Perturbations as in (27) could generate a discontinuous gradient flow as in Example 4.2, or non-trivial perturbed minimizing movements in cases in which classical minimizing movements are only the constant motion. Hence, perturbed minimizing movements can be used to obtain motion from a setting which does not allow it for classical minimizing movements, as we will see in the next section.
and initial data u τ 0 ≡ 0. Assumptions H1-H4 hold. If we consider perturbations satisfying assumption H5, by Theorem 2.4, we obtain continuous perturbed minimizing movements, but the only continuous curve on a discrete set (the domain of the energy is Z) is the constant curve. Therefore every continuous perturbed minimizing movement for this problem, in particular the classical minimizing movement, is the trivial motion.
Given any T, δ > 0, we consider a τ as in (27) satisfying H5 ' with I = T N and δ(τ ) = δτ . First, note that the minimum of the function t → φ(t)+a τ n (t−u τ n−1 ) 2 /2τ is the integer nearest to the minimum of the continuous function t → −t + a τ n (t − u τ n−1 ) 2 /2τ . Let t τ n be such a minimum, we have that t τ n = u τ n−1 + τ /a τ n . Hence the n-th step of the discrete solution of the scheme (2) is u τ n = t τ n + 1 2 = u τ n−1 + τ a τ n + 1 2 .
For the sake of simplicity, we ignore the bifurcation phenomenon (which could be treated separately); hence we consider δ such that 1/δ ∈ N + 1/2. We define N := 1/δ + 1/2 . If n = T k/τ for some integer k, we get u τ n = u τ n−1 + N , otherwise u τ n = u τ n−1 . Hence the discrete solution is Taking the limit as τ → 0, we obtain the perturbed minimizing movement u(t) = N t/T , which is not the trivial motion when δ < 2.
The argument of the previous example can be iterated when a multi-well quadratic energy functional is considered. While perturbed minimizing movements corresponding to perturbations satisfying assumption H5, in particular classical minimizing movements, would stagnate in the initial potential well, perturbed minimizing movements with perturbations as in H5 ' may explore different local minima.
We study the minimum of the function t → φ(t) +δ(t −ū) 2 /2 using calculations done in Example 4.5 by changing variable t in t − k or t − (k − 1), and adding k or k − 1 respectively which does not affect the minimization. With same notation we get Hence, by comparing the minima, for any k > 0 we obtain that Note that this is not in contrast with the previous condition onū. From the previous computation, we have that the minimizer of the map t → φ(t)+δ(t−ū) 2 /2 is t k (δ,ū) whenever we haveū ∈ (k − 1 − 1/δ, k − 1/δ]. The behavior of the motion depends on δ. Fixed T > 0 sufficiently large, we consider discrete steps u τ n such that n/τ < T . First, we assume that δ ≥ 1. Let the discrete solution u τ be in the interval (h − 1, h], that is the h-th energy well. As in condition (28) we have that, if u τ n−1 ∈ (h − 1, h − 1 δ ], the discrete solution will not pass to another potential well, when τ is small enough; whereas, if it is in (h − 1 δ , h], it will pass to the next well; i.e, u τ n ∈ (h, h + 1], provided that n is such that a τ n = δτ . Figure 6. Graphs for δ = 8. As in Figure 5, the motion does not exit the well when t = 1, but when t = 2 it does.
As pictured in Figure 7, if δ = 1 the motion will always exit any potential well as soon as a τ n = τ , because the range of the positions of u τ n−1 , in respect of which u τ n pass to the next well, is the whole interval (h−1, h]; hence the perturbed minimizing movement u(t) ∈ (h − 1, h) whenever t ∈ (h, h + 1), for every positive integer h. If 0 < δ < 1, the discrete solution will exit more than one well. If u τ n−1 ∈ (h − 1, h − 1/δ], it will pass through 1/δ wells, if (h − 1/δ, h] through 1/δ ; i.e., u τ n ∈ (h − 1 + 1/δ , h + 1/δ ], or u τ n ∈ (h − 1 + 1/δ , h + 1/δ ] respectively. Figure 8. Graphs of a discrete solution passing through two potential wells at every jump discontinuity. We conclude this paper studying a particular case in which the perturbations do not even satisfy the relaxed assumption H5 ' . The following result is stated in a restrictive situation and could be generalized, but the aim is to show that, even for uncontrolled perturbations, discrete solutions may converge to a perturbed minimizing movement. Proposition 7. Let φ satisfy assumptions H1-H3, and assume that it admits a unique global minimum u * . Let u τ 0 satisfy H4, and let {a τ } be a family of perturbations as in (1). If there exists a time parameter t 0 > 0 such that (i) a τ (t) satisfies assumption H5 ' for every t ∈ [0, t 0 /τ τ ] (ii) lim τ →0 a τ t0/τ /τ = 0, then there exists a {a τ }-perturbed minimizing movement u such that u(t) = u * for every t > t 0 .