Minimizing movements along a sequence of functionals and curves of maximal slope

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1 Minimizing movement along a equence of functional and curve of maximal lope Andrea Braide Dipartimento di Matematica, Univerità di Roma Tor Vergata via della ricerca cientifica 1, 133 Roma, Italy Maria Colombo ETH Intitute for Theoretical Studie Clauiutrae Zürich, Switzerland Maimo Gobbino Dipartimento di Matematica, Univerità di Pia via Filippo Buonarroti 1c, Pia, Italy Margherita Solci DADU, Univerità di Saari piazza Duomo 6, 741 Alghero (SS), Italy Abtract We prove that a general condition introduced by Colombo and Gobbino to tudy it of curve of maximal lope allow alo to characterize minimizing movement along a equence of functional a curve of maximal lope of a it functional. 1 Introduction Following a vat earlier literature, the notion of minimizing movement ha been introduced by De Giorgi to give a general framework for Euler cheme in order to define a gradient-flow type motion alo for a non-differentiable energy φ. It conit in introducing a time cale τ, define a time-dicrete motion by a iterative minimization procedure in which the ditance from the previou tep i penalized in a way depending on τ, and then obtain a timecontinuou it a τ. Thi notion i at the bae of modern definition of variational motion and ha been uccefully ued to contruct a theory of gradient flow in metric pace by Ambroio, Gigli and Savaré [3]. In particular, under uitable aumption it can be hown that a minimizing movement i a curve of maximal lope for φ. When a equence of energie φ ε parameterized by a (mall poitive) parameter ε ha to be taken into account, in order to define an effective motion one may examine the minimizing movement of φ ε and take their it a ε, or, depending on the problem at hand, intead compute the minimizing movement of the (Γ-)it φ of φ ε. In general 1

2 thee two motion are different. Thi i due to the trivial fact that the energy landcape of φ may not carry enough information to decribe the energy landcape of φ ε, ince local minimizer may appear or diappear in the it proce, a eay example how [4]. A general approach i to proceed in the minimizing-movement cheme letting the parameter ε and the time cale τ tend to together. Thi give a notion of minimizing movement along φ ε at a given cale τ. In thi way we can detect fine phenomena due to the preence of local minima. The it of the minimizing movement and the minimizing movement of the it are recovered a extreme cae. A firt example of thi approach ha been given in [6] for pin energie converging to a crytalline perimeter, in which cae the extreme behaviour are complete pinning and flat flow [1]. For a general choice of the parameter the it motion i neither of the two but depend on the ratio between ε and τ and i a degenerate motion by crytalline curvature with pinning only of large et. More example can be found in [4]. Note that in general the function that we obtain a minimizing movement along a equence cannot be eaily rewritten a minimizing movement of a ingle functional. In another direction, condition have been exhibited that enure that the it of gradient flow for a family φ ε, or of curve of maximal lope, be the gradient flow, or a curve of maximal lope, for their Γ-it φ (ee [8] and [7]). Thi ugget that under uch condition all minimizing movement along φ ε, at whatever cale, may converge to minimizing movement of φ. In thi paper we prove a reult in that direction, howing that if a lower-emicontinuity inequality hold for the decending lope of φ ε then any minimizing movement i a curve of maximal lope for φ (Theorem 4). Thi property hold in particular in the cae of convex energie (ee [7], and alo [2] and [4]). Note that the it curve of maximal lope may till depend on the way ε and τ tend to, and that in the extreme cae of ε tending to fat enough with repect to τ it i alo a minimizing movement for the it (Theorem 8(b)). In the cae that all curve of maximal lope are minimizing movement for φ, thi i a kind of commutativity reult between minimizing movement and Γ-convergence. Thi again hold if φ i convex, which i automatic if alo φ ε are convex. Neverthele, in ome cae it ha been poible to directly prove that minimizing movement along a equence are minimizing movement for the it, alo for ome non-convex energie a caled Lennard-Jone one [5]. The reult preented in thi note i uggeted by the analog reult for curve of maximal lope in [7], it make the analyi in [4] more precie, and we think it will be a ueful reference for future application. It proof follow modifying the argument of [3], which how that the minimizing movement for a ingle functional are curve of maximal lope, and i briefly preented in Section 3. 2 The it reult In what follow (X, d) i a complete metric pace. Definition 1 (Minimizing movement along φ ε at cale τ ε). For all ε > let φ ε : X (, + ], and let u ε X. Suppoe that there exit ome τ > uch that for every ε > and τ (, τ ), there exit a equence {u i ε,τ } which atifie u ε,τ = u ε and u i+1 ε,τ i a olution of the minimum problem n min φ ε(v) + 1 o 2τ d2 (v, u i ε,τ ) : v X. (1) Let τ = τ ε be a family of poitive number uch that ε τ ε = and define the piecewiecontant function u ε = u ε,τε : [, + ) X a u ε(t) = u i+1 ε,τ for t (iτ, (i + 1)τ]. A 2

3 minimizing movement along φ ε at cale τ ε with initial data u ε i any pointwie it of a ubequence of the family u ε. From now on, we will make the following hypothee, which enure the exitence of minimizing movement along the equence φ ε at any given cale τ ε [4]: (u X i an arbitrary given point) (i) for all ε > φ ε i lower emicontinuou; (ii) there exit C > and τ > uch that inf φ ε(v) + 1 d(v, u ) : v X C > 2τ for all ε > ; (iii) for all C > there exit a compact et K uch that {u : d 2 (u, u ) C, φ ε(u) C} K for all ε >. Remark 2. (a) Alternatively, in the definition above we can uppoe τ and chooe ε = ε τ. In thi way we define minimizing movement along φ ετ at cale τ; (b) If φ ε = φ for all ε, then a minimizing movement along φ ε at any cale i a (generalized) minimizing movement for φ a defined in [3]. For φ: X (, + ] we define the (decending) lope of φ a 8 (φ(x) φ(y)) + >< up if φ(x) < + and x i not iolated y x d(x, y) φ (x) = if φ(x) < + and x i iolated >: + if φ(x) = +. We ay that v : [, T ] X belong to AC 2 ([, T ]; X) if there exit A L 2 (, T ) uch that d(v(), v(t)) (2) A(r) dr for any t T. (3) The mallet uch A i the metric derivative of v and it i denoted by v. Definition 3 (Curve of maximal lope). A curve of maximal lope for φ: X (, + ] in [, T ] i u AC 2 ([, T ]; X) uch that there exit a non-increaing function ϕ: [, T ] R uch that ϕ() ϕ(t) 1 2 u 2 (r) dr φ 2 (u(r)) dr for any t T (4) and a Lebegue-negligible et E uch that φ(u(t)) = ϕ(t) in [, T ] \ E. Theorem 4 (Minimizing movement and curve of maximal lope). Let φ ε, φ: X (, + ] atify (i)-(iii) and the following condition: (H) for all ubequence φ εn and v n v with up n { φ εn (v n) + φ εn (v n)} < +, we have φεn (v n) = φ(v) and inf φεn (v n) φ (v). (5) Let u ε,τ be uch that there exit S, S for which d 2 (u ε,τ, u ) S < + and φ ε(u ε,τ ) S < +. Then for all τ ε any minimizing movement along φ ε at cale τ ε with initial data u ε,τ ε i a curve of maximal lope for φ. Remark 5. (a) Note that we do not uppoe that φ ε Γ-converge to φ. Thi i uually deduced at point with finite lope for φ; (b) The reult can be proved under the more general aumption that hypothee (i) and (iii) hold with repect to a weaker topology compatible with the ditance d (ee [3, Sect. 2.1]); (c) If the initial lope atify the additional bound φ ε (u ε,τ ) S < +, then any it curve u a in the tatement of Theorem 4 atifie φ(u(t)) φ(u()) for almot all t T. 3

4 Corollary 6 (Curve of maximal lope and Γ-convergence). Let φ ε atify (i)-(iii) and φ ε(y) φ ε(x) d(x, y) φ ε (x) for any y X (6) for any x uch that φ ε(x) and φ ε (x) are finite (Slope Cone Property [7]). Let φ ε Γ- converge to φ with repect to d. Then the claim of Theorem 4 hold with φ = φ. The proof of the corollary follow from [7, Propoition 3.4]. Note that Corollary 6 hold even if aumption (6) i weakened by ubtracting in the right-hand ide a lower-order term with repect to d(x, y), a decribed in [7, Remark 3.5]. Remark 7 (Convex energie). If φ ε are convex, then they atify condition (6). We note that in general the claim of Theorem 4 doe not hold under the only hypothei of Γ-convergence of φ ε. More preciely, the following theorem give a connection between minimizing movement along a equence and the (generalized) minimizing movement of the it. Theorem 8 (Γ-convergence and minimizing movement). Let φ ε atify (i)-(iii). Then (a) there exit τ = τ ε uch that if τ ε τ ε then each minimizing movement along φ ε at cale τ ε i (up to ubequence) the it a ε of the (generalized) minimizing movement for φ ε; (b) if φ ε Γ-converge to φ with repect to d, then there exit ε = ε τ uch that if ε τ ε τ then each minimizing movement along φ ετ at cale τ i (up to ubequence) a (generalized) minimizing movement for φ. In [4, Theorem 8.1] it i proved the exitence of familie τ ε and ε τ uch that the concluion of (a) and (b) hold for minimizing movement along φ ε exactly at cale τ ε and for minimizing movement exactly along φ ετ at cale τ repectively. Theorem 8 follow by noticing that the argument of that proof imply that we may define τ ε and ε τ uch that thi hold for τ ε τ ε and ε τ ε τ, repectively. In cae (b), note that if the it φ atifie ome additional differentiability hypothee (ee [3, Theorem and 2.3.3]) then each minimizing movement i a curve of maximal lope for the it φ. 3 Proof of Theorem 4 In order to prove a priori uniform etimate we introduce ome definition in analogy to thoe in [3, Section 3.2]. For any ε > and δ (, τ ), given u X we denote by J ε,δ (u) the et of v minimizing φ ε( ) + 1 2δ d2 (, u). Then ũ ε,τ : [, + ) X i any interpolation of the value of u ε,τ in τn atifying ũ ε,τ (t) J ε,t iτ (u i ε,τ ) for t (iτ, (i + 1)τ], and Note that for t > G ε,τ (t) = up{d(v, ui ε,τ ) : v J ε,t iτ (u i ε,τ )} t iτ for t (iτ, (i + 1)τ]. (7) G ε,τ (t) φ ε (ũ ε,τ (t)). (8) Furthermore, u ε,τ denote the piecewie-contant function defined by u ε,τ (t) = τ 1 d(u i+1 ε,τ, u i ε,τ ) for t (iτ, (i + 1)τ). The following lemma i the analog of [3, Lemma 3.2.2] for a equence φ ε. 4

5 Lemma 9 (A priori uniform etimate). Let ε, τ > and τ < τ. Let φ ε atify (i) (iii), and u ε,τ be uch that there exit S, S for which d 2 (u ε,τ, u ) S < + and φ ε(u ε,τ ) S < +. Let {u i ε,τ } i be defined a in Definition 1 with initial data u ε,τ. Then for any i, j N with i < j φ ε(u i ε,τ ) φ ε(u j ε,τ ) = 1 2 Z jτ iτ u ε,τ 2 dt Z jτ iτ G ε,τ 2 dt. (9) Moreover, fixed T >, there exit a contant C depending only on T, S, S, τ, u uch that, for τ < τ /8, for any N with Nτ T d 2 (u N ε,τ, u ) C, φ ε(u N ε,τ ) C (1) d 2 (ũ ε,τ (t), u ε,τ (t)) Cτ in [, T ] (11) Z Nτ Z Nτ u ε,τ 2 (r) dr φ ε(u ε) φ ε(u N ε,τ ) C (12) G ε,τ 2 (r) dr φ ε(u ε) φ ε(u N ε,τ ) C. (13) The proof follow that of [3, Lemma 3.2.2], where a ingle φ i conidered. The uniform bound are enured by the condition (ii) for the equence φ ε and by the uniform bound on the initial data, which allow to prove that for any i d 2 (u i ε,τ, u ) 2S + 2τ S 2τ C + 4 τ ix τ d 2 (u j ε,τ, u ). (14) An application of the Gronwall Lemma to thi inequality give the required uniform bound (ee [3, pp ]). With the aid of thi lemma we can prove Theorem 4 by howing that, up to ubequence, v n = ũ εn,τ εn (t) atifie the hypothee of condition (H) for almot all t. Etimate (12) implie the weak convergence in L 2 loc[, + ) of a ubequence u ε n,τ n A. Since each φ εn (ũ εn,τ n ( )) i not increaing (ee [3, Lemma 3.1.2]), we may apply Helly Lemma (e.g. [3, Lemma 3.3.3]) obtaining (up to a further ubequence) ϕ(t) = j=1 φεn (ũ εn,τn (t)) for any t [, T ]; (15) in particular, the family φ εn (ũ εn,τn (t)) i equibounded for any t [, T ]. From (8), etimate (13) and Fatou Lemma Z T «Z T inf φεn (ũ εn,τn (t)) dt inf φ εn 2 (ũ εn,τn (t)) dt < +, we get that there exit a Lebegue-negligible et E uch that inf φεn (ũ εn,τ n (t)) < + t [, T ] \ E. (16) The etimate in (1) and the compactne hypothei (iii) imply that the family u εn,τn i contained in a compact ubet of X. We et (n) = i nτ n with (n) and (n), and t(n) = j nτ n with t(n) t and t(n) t. By the definition of u ε n,τ n and by weak convergence we get up d(u εn,τ n (), u εn,τ n (t)) up (n) (n) u ε n,τ n (r) dr A(r) dr (17) 5

6 for any, t [, T ]. Thi etimate and the pre-compactne of the family u εn,τn allow to ue a Acoli-Arzelà argument (ee e.g. [3, Propoition 3.3.1]) to obtain the exitence of a function u pointwie it of a further ubequence of u εn,τn (t). Since (11) hold, alo ũ εn,τn (t) converge to u(t) for any t [, T ]. Since (16) i in force, we can extract for any t [, T ] \ E a (t-dependent) equence n k + uch that k + φεn k (ũεn k,τn (t)) = inf k φεn (ũ εn,τn (t)) < +. Applying (H) to ũ εnk,τ nk (t), for t [, T ] \ E we get φ(u(t)) = φ (u(t)) k + φεn k,τn k (ũεn k,τn (t)) = k φεn,τn (ũ εn,τn (t)) = ϕ(t) (18) k + φεn k (ũεn k,τn (t)) = inf k φεn (ũ εn,τn (t)). (19) It remain to prove that u i a curve of maximal lope for φ. Paing to the it in (17) it follow that u AC 2 ([, T ]; X) and that for almot all t [, T ] u (t) A(t). The weak convergence enure that inf (n) (n) u ε n,τ n 2 (r) dr A 2 (r) dr Etimate (8) and (19), and an application of Fatou Lemma imply inf (n) (n) G εn,τ n 2 (r) dr inf (n) (n) φ εn 2 (ũ εn,τ n (r)) dr u 2 (r) dr. (2) φ 2 (u(r)) dr. (21) By the non-increaing monotonicity of φ εn (ũ εn,τn ( )) we have the inequalitie φ εn (u in ε n,τ n ) φ εn (ũ εn,τn ()) and φ εn (u jn ε n,τ n ) φ εn (ũ εn,τn (t)), o that, alo uing (2), (21) and (18), from (9) we obtain ϕ() ϕ(t) 1 2 where φ(u(t)) = ϕ(t) in [, T ] \ E. Acknowledgement u 2 dr φ 2 (u(r)) dr, (22) The author thank the anonymou referee for her/hi ueful comment. The econd author ha been partially upported by Dr. Max Röler, the Walter Haefner Foundation and the ETH Zurich Foundation. Reference [1] F. Almgren, J.E. Taylor. Flat flow i motion by crytalline curvature for curve with crytalline energie. J. Differential Geom. 42 (1995), [2] L. Ambroio and N. Gigli. A uer guide to optimal tranport, in Modelling and Optimiation of Flow on Network (B. Piccoli and M. Racle ed.) Lecture Note in Mathematic. Springer, Berlin, 213, pp [3] L. Ambroio, N. Gigli and G. Savaré. Gradient Flow in Metric Space and in the Space of Probability Meaure. Lecture in Mathematic ETH, Zürich. Birkhhäuer, Bael, 28. 6

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