Hidden convexity in a problem of nonlinear elasticity
Nassif Ghoussoub, Young-Heon Kim, Hugo Lavenant, Aaron Zeff Palmer
aa r X i v : . [ m a t h . A P ] A p r HIDDEN CONVEXITY IN A PROBLEM OF NONLINEAR ELASTICITY
NASSIF GHOUSSOUB, YOUNG-HEON KIM, HUGO LAVENANT, AND AARON ZEFF PALMER
Abstract.
We study compressible and incompressible nonlinear elasticity variational problems in a gen-eral context. Our main result gives a sufficient condition for an equilibrium to be a global energy minimizer,in terms of convexity properties of the pressure in the deformed configuration. We also provide a convexrelaxation of the problem together with its dual formulation, based on measure-valued mappings, whichcoincides with the original problem under our condition.
Contents
1. Introduction 12. Existence of minimizers and optimality conditions 33. Global optimality 74. Examples 95. A convex relaxation 10Acknowledgments 15References 151.
Introduction
In this article, we study a problem of calculus of variations of the form(1.1) min u " ˆ Ω ´ W p ∇ u q ` F p u q ¯ d L Ω ` Φ p u L Ω q ; u : Ω Ñ D and u “ g on B Ω * where the unknown, u : Ω Ă R d Ñ D Ă R k , is a vector-valued function representing the deformation ofan elastic solid when d “ k . The data consists of the boundary values fixed by a function g : B Ω Ñ B D ;a potential energy F : Ω ˆ D Ñ R ; and the hyper-elastic stored energy W : Ω ˆ R k ˆ d Ñ R (dependenceon x P Ω is suppressed in our notation for (1.1) and what follows). The functional Φ is convex on theset of finite nonnegative measures over D . We let L Ω denote the Lebesgue measure restricted to Ω and u L Ω is the push-forward image measure of L Ω by u .The first example is the incompressibility constraint, imposed byΦ inc p µ q “ µ “ L D `8 otherwise . When d “ k , any u with u L Ω “ L D will be called incompressible, and if such a u is sufficientlydifferentiable and injective then equivalently the Jacobian equation det p ∇ u q “ Department of Mathematics, University of British Columbia
E-mail addresses : {nassif,yhkim,lavenant,azp}@math.ubc.ca . Date : April 23, 2020.2010
Mathematics Subject Classification.
Primary 74G65. Secondary 49N15, 35Q74. also consider integral function of the density, that is(1.2) Φ p µ q “ $&% ˆ D φ ˆ d µ d L D ˙ d L D if µ ! L D `8 otherwise . where φ : r , `8q Ñ R is a convex function. As shown in Section 1.2, this will correspond to the energyof compressible deformations in nonlinear elasticity.The presence of Φ composed with the image measure of u makes problem (1.1) nonlinear and non-convex. However, we will provide a sufficient condition for a solution of the Euler-Lagrange equation of (1.1) to be the unique global minimizer of (1.1). Moreover, we will see that under this assumption theproblem in fact coincides with a convex relaxation built on measure valued mappings.First, let us motivate this study. Though we will mainly deal with examples where d “ k , that is Ωand D lie in the same Euclidean space, our results also apply to d ‰ k . In particular, the case k “ Optimal transport.
If Ω “ D and W “ “ Φ inc and F p x, u q “ ´ f p x q ¨ u , then the problem readsmax u " ˆ Ω f ¨ u d L Ω ; u L Ω “ L Ω * , and one recovers the problem of polar factorization studied by Brenier [6, 8]. In this case, provided f L Ω does not charge small sets, then f can be uniquely written f “ p ∇ ω q ˝ u where ω is convex and u is asolution of the problem. Actually, ∇ ω is the unique gradient of a convex function mapping f L Ω onto L Ω .Hence Problem (1.1) can be seen as an optimal transport problem to which one adds a gradientpenalization. Such problems have been considered in the PhD thesis of Louet [23] which in particularstudies in great details the existence of a solution. What we have here can be seen as a simplified versionof their problem: we only look at Φ p u L Ω q while they consider Φ p u α q with α P M p Ω q possibly singular.However, the result that we have (solutions of the Euler-Lagrange equations are global minimizers underappropriate conditions) was not addressed at all in their work.1.2. Non Linear elasticity.
In nonlinear elasticity theory, one is led to consider problem of the form(1.3) min u " ˆ Ω ´ W p ∇ u q ` h p det ∇ u q ` F p u q ¯ d L Ω ; u “ g on B Ω * , where k “ d and u : Ω Ñ D is the deformation of an elastic solid. The functions W and h are assumedto be convex, in such a way that C ÞÑ W p C q ` h p det C q is a polyconvex function on the set of matrices.The function h p t q usually tends to `8 when t Ñ
0, and a limit case is when h p t q “ `8 for all t but t “
1, which implies that u L Ω “ L D and u is incompressible.By a change of variable formula, this problem falls into our framework. Let us define the function φ h by φ h p s q “ h p s ´ q s and the functionalΦ h p µ q “ $&% ˆ D φ h ˆ d µ d L D ˙ d L D if µ ! L D `8 otherwise . which we already introduced in (1.2). Then, for injective u we can write ˆ Ω h p det ∇ u q d L Ω “ Φ h p u L Ω q . IDDEN CONVEXITY IN A PROBLEM OF NONLINEAR ELASTICITY 3
To justify this, we can note that d L D d u L Ω ˝ u “ det p ∇ u q and calculateΦ h p u L Ω q “ ˆ D φ h ˆ d u L Ω d L D ˙ d L D “ ˆ D h ˆ d L D d u L Ω ˙ d u L Ω “ ˆ Ω h p det ∇ u q d L Ω . The function φ h is convex (hence the functional Φ h is also convex on the set of measures) provided h isconvex, since we can write it as a supremum of convex functions: φ h p s q “ sup r p r ´ sh ˚ p r qq (with h ˚ being the Legendre transform of h ). The now classical work of Ball [2], guarantees conditionson W and h such that minimizers exist in a class of weakly differentiable Sobolev functions. Furtherregularity of these minimizers remains unknown as a major unsolved problem.One may argue that prescribing the codomain D is too restrictive. However, we emphasize that westudy only Dirichlet boundary conditions g hence the domain D could be defined implicitly as the onesuch that B D “ g pB Ω q .The question of uniqueness of solutions to the equilibrium equations, as well as whether they areglobal minimizers is an open question in calculus of variations [4, 5], with counterexamples for somespecial cases. Examples inspired from the theory of elasticity will be discussed in Section 4. We canalready highlight some insight from this theory: ‚ To get uniqueness of equilibrium, one needs to restrict to pure displacement on the boundaries,that is, only Dirichlet boundary conditions as we have done here. ‚ There are situations where multiple equilibrium solutions exist which are all global minimizersbut which are not continuous [3]. Superlinear growth of C ÞÑ W p C q or stronger seems necessaryto avoid cavitation and guarantee uniqueness. ‚ If the domain Ω is not simply connected, it may be possible to construct examples with infinitelymany local minima (hence solutions of the equilibrium equations) but only one global minimizer[24].Our main result is that any solution of the equilibrium equations which is smooth and small in a certainsense (namely the pressure, expressed in the deformed configuration must be λ -convex with λ not toonegative) is the unique global minimizer (but there may be other local minima). It can be seen as apartial answer to [4, Problem 8].In the rest of this article, we first state rigorously the problem and derive the Euler-Lagrange equations.Then we state our sufficiency condition for being an global minimizer and we prove it. We provideexamples to discuss the sharpness of the result. Eventually we introduce a convex relaxation based onmeasure valued mappings which coincides with the original problem under the same smallness assumptionon the pressure. This last part does not provide new results on the original problem but we think isinteresting on its own and relates to the parallel work of [1], which was the starting point of our study.2. Existence of minimizers and optimality conditions
Notation and the variational problem of interest.
Let Ω Ă R d and D Ă R k be open andbounded domains with Lipschitz boundaries. We denote by respectively L Ω and L D the Lebesgue measurerestricted to Ω and D . The set of finite Radon measures on a metric space X is denoted by M p X q and weendow it with the topology of weak* convergence. The set of positive Radon measures on X is denotedby M ` p X q .If T : X Ñ Y and α P M p X q we denote by T α P M p Y q the push-forward of the measure α by T ,that is the measure defined by T α p B q “ α p T ´ p B qq for any Borel set B in Y . We often make use of HIDDEN CONVEXITY IN A PROBLEM OF NONLINEAR ELASTICITY the change of variables, that if T : X Ñ Y is α measureable and g is continuous on Y then ˆ X g ˝ T d α “ ˆ Y g d T α. The notation | x | stands for the Euclidean norm of x if x is a vector, and the Frobenius norm (a.k.aHilbert-Schmidt norm) if x is a matrix.For the data of the problem, let W : Ω ˆ R k ˆ d Ñ R be a Carathéodory function satisfying the coercivityand boundedness assumptions 1 C p| H | p ´ q ď W p H q ď C p| H | p ` q for some C ą p ą
1, and we assume that H ÞÑ W p H q is convex almost everywhere on Ω. We take g : B Ω Ñ B D for the boundary condition, which we assume extends to a function on Ω with g p Ω q “ D ,and F a Carathéodory function satisfying for all u : Ω Ñ D , ˆ Ω | F p u q| d L Ω ă `8 , and u ÞÑ F p u q is convex. Remark . Not to overburden notations, we suppress the dependence on x of W and F in the presentarticle. For instance, an expression like W p ∇ u q means W p x, ∇ u p x qq while F p u q “ F p x, u p x qq .We denote by W ,pg p Ω , D q the set of Sobolev functions u P W ,p p Ω , R k q such that the trace of u on B Ωis g and the range of u remains in D . We let ¯ D “ D Y B D and we take Φ : M ` p ¯ D q Ñ R Y t`8u to beconvex, bounded below and lower semi-continuous.
Definition 2.2.
We define the energy E : W ,p p Ω , R k q Ñ R Y t`8u as E p u q “ ˆ Ω ´ W p ∇ u q ` F p u q ¯ d L Ω ` Φ p u L Ω q . The problem with penalization of the image measure we are looking at is(2.1) inf E p u q ; u P W ,pg p Ω , D q ( . As noted in Proposition 2.3 below, it is also possible to include the additional constraint that u isinjective on Ω.Note that is is not obvious that there exists at least one admissible competitor in (2.1): existence ofsmooth u satisfying the constraint u α “ β has been investigated with positive answers for Lebesguemeasure in [13] and in [14, Theorem 1.1] (but without boundary conditions for u ) provided the measures α and β satisfy smoothness assumptions. In the compressible case of Φ h , existence is easier to get, butone still has to control that det ∇ u from above and below. In any case, once existence of a competitor isknown then existence of a minimizer of E follows easily. Proposition 2.3.
Assume for p ą that there exists u P W ,pg p Ω , D q such that Φ p u L Ω q ă `8 . Thenthere exists a global minimizer of E in W ,pg p Ω , D q . If p ą d , then the result holds with the additionalconstraint that u is injective.Proof. It comes from the direct method of calculus of variations. As Φ is bounded from below, with thecoercivity of W it is easy to show that any minimizing sequence p u n q n P N is bounded in W ,p p Ω , R k q . Upto extraction of a subsequence, p u n q n P N converges to u weakly in W ,p p Ω , R k q , strongly in L p p Ω , R k q andalmost everywhere. The latter convergence implies that lim n u n L Ω “ u L Ω in the weak* topology.Then all the terms of E are lower semi-continuous so the limit u is a global minimizer. In the casethat we include the constraint that u is injective, weak compactness still holds for injective functions in W ,pg p Ω , D q when p ą d using the result of [12]. (cid:3) IDDEN CONVEXITY IN A PROBLEM OF NONLINEAR ELASTICITY 5
We now wish to further compare our result with the classical results of nonlinear elasticity. Indeed, theproblem (2.1) is equivalent to the nonlinear elasticity problem (1.3) when p ą d and Φ has the form of Φ inc or Φ h . We consider a convex, lower semicontinuous function h : R Ñ R Y t`8u such lim t Ñ h p t q “ `8 and h is bounded from below, for which Φ h is convex, lower semicontinuous and bounded below. Lemma 2.4.
We suppose that p ą d “ k and g is orientation preserving. Let u P W ,pg p Ω , D q one toone. Then Φ inc p u L Ω q ă `8 if and only if det ∇ u “ holds almost everywhere on Ω . Similarly, thereholds ˆ Ω h p det ∇ u q d L Ω “ Φ h p u L Ω q , where in particular the left hand side is finite if and only if the right hand side is.Proof. This lemma follows immediately from the fact that if u P W ,p p Ω , R k q and u is almost everywhereinjective, then d u L Ω d L D “ | det p ∇ u q| ´ ˝ u ´ . If g is orientation preserving, this implies that det p ∇ u q ą (cid:3) Optimality conditions.
Now we turn to the derivation of the optimality condition. To that extent,we will rely on the notion of the subdifferential of Φ in the sense of convex analysis.For any ω P C p ¯ D q , we define the Legendre transform of Φ asΦ ˚ p ω q “ sup µ P M p ¯ D q ˆ D ω d µ ´ Φ p µ q . We say that ω P C p ¯ D q belongs to the subdifferential of Φ at µ , and we write ω P B Φ p µ q , if Φ ˚ p ω q “ ´ ω d µ ´ Φ p µ q . Example 2.5.
In the case where Φ h p µ q “ ˆ D φ h ˆ d µ d L D ˙ d L D with φ h p s q “ h p s ´ q s for a smooth and convex h , and if µ has a smooth density w.r.t. L D then B Φ h p µ q consists of the single element ω “ φ h ˆ d µ d L D ˙ . Written in terms of u and h , if u P C p ¯Ω , ¯ D q smooth and invertible, then the subdifferential B Φ h p u L Ω q consists of the single element ω “ ` ´ h p det ∇ u q det p ∇ u q ` h p det ∇ u q ˘ ˝ u ´ . Example 2.6.
As another important case, let Φ inc be defined byΦ inc p µ q “ µ “ L D `8 otherwise , then Φ ˚ inc p ω q “ ´ D ω d L D and any ω P C p ¯ D q belongs to B Φ inc p L D q .Now we claim that the strong form of the Euler Lagrange equations may be expressed as(2.2) ∇ ¨ DW p ∇ u q ´ DF p u q “ p ∇ ω q ˝ uω P B Φ p u L Ω q , where DW and DF represent the differential with respect to their second argument, that is the one in R dk and R k respectively. Note that B Φ p u L Ω q ‰ H implies that Φ p u L Ω q ă `8 . HIDDEN CONVEXITY IN A PROBLEM OF NONLINEAR ELASTICITY
Definition 2.7.
Provided that ∇ ω is well defined, we say that (2.2) holds weakly if for all v P W ,p p Ω , R k q , ˆ Ω ´ ∇ v : DW p ∇ u q ` v ¨ ` DF p u q ` p ∇ ω q ˝ u ˘¯ d L Ω “ . Later we will consider the case ω is λ -convex where ∇ ω will be defined as a measurable selection of thesubdifferential of ω .In the case of Φ “ Φ h for ω P B Φ p u L Ω q we have p ∇ ω q ˝ u “ ´ det p ∇ u q ∇ u ´J ∇ h p det ∇ u q , which agrees with what one obtains through the direct variation of u with integration by parts and theidentity that ∇ ¨ det p ∇ u q ∇ u ´J “ Remark . In elasticity theory the equilibirum equations are not usually written in this way. Indeedif we introduce p “ ω ˝ u and S “ DW p ∇ u q the second Piola-Kirchoff stress tensor then (2.2) can bewritten ∇ ¨ S ´ DF p u q “ ∇ u ´J ∇ p on Ω . Alternatively, we introduce the Cauchy stress T “ S p ∇ u q J ˝ u ´ , in which case the equilibrium equationsare simply(2.3) ∇ ¨ T ´ DF p u q ˝ u ´ “ ∇ ω on D. The assumption of our main theorem will be about ω , that is about the pressure in deformed configuration .Let us justify that (2.2) are indeed the Euler Lagrange equations in the case where the solution issmooth. Indeed, the existence of a pressure is guaranteed when u is sufficiently smooth, following thearguments of [22]. Results can also be attained for small forces as in [21]. In some cases this argumentcan be weakened to obtain a distributional solution; see the arguments of [15]. Applying the methods ofthese articles we can obtain the following. Here we do not aim for the sharpest regularity assumptionswhich may depend on the particular form of the functional Φ.By common abuse of notation, in the proposition below and its proof we identify a measure on D withits density w.r.t. L D . Moreover, we will say that Φ : M ` p ¯ D q Ñ R Y t`8u is regular if it is a convex,bounded from below, lower semi continuous function which coincides with its lower semi continuousenvelope of when restricted to measures with a smooth density. Specifically, for every µ P M ` p ¯ D q withΦ p µ q ă `8 , we assume that there exists a sequence p µ n q n P N of measures in W j,r p D q with jr ą d suchthat Φ p µ n q converges to Φ p µ q when n Ñ `8 . If Φ is either Φ inc or Φ h for a smooth and convex h thenit is clearly regular. Proposition 2.9.
Assume that Φ is regular. We suppose that B Ω , B D, g, W, F are smooth, and u P W ,pg p Ω , D q is a local minimizer of E p u q such that ă u L Ω P W j ` ,r p D q , the Cauchy stress satisfies T P W j,r p D, R dk q , DF p u q ˝ u ´ P W j ´ ,r p D, R k q for j P N , ă r ă `8 , and rj ą d . Then there exists ω P W j,r p D q X B Φ p u L Ω q such that (2.2) holds weakly and (2.3) holds strongly on D .Proof. Let µ “ u L Ω , in particular µ P W j ` ,r p D q .We start by proving the existence of the pressure ω . We consider variations of the form u ` v ˝ u for v : D Ñ R k with v “ B D . We define G : v ÞÑ p u ` v ˝ u q L D . We let˜ E p v q “ ˆ Ω ` W p ∇ p u ` v ˝ u qq ` F p u ` v ˝ u q ˘ d L Ω . The problem to minimize ˜ E p v q subject to G p v q “ µ has a local minimum at v “
0. Linearizing theconstraint, we have for z P C p ¯ D q , x z, DG p v q q y “ dd t ˆ D z d p u ` t q ˝ u q L Ω ˇˇˇ t “ “ ddt ˆ D z ˝ p¨ ` t q q d µ ˇˇˇ t “ “ ˆ D ∇ z ¨ q d µ. IDDEN CONVEXITY IN A PROBLEM OF NONLINEAR ELASTICITY 7
The map DG p v q : W j ` ,r p D q Ñ W j,r p D q , which can be expressed as DG p v q q “ ´ ∇ ¨ p q µ q , is continuous and has closed range as a composition of the divergence operator and multiplication bya positive function in W j ` ,r p D q . Thus by the Lagrange multiplier theorem there exists ω P W ´ j,r p D q with ´ D ω d µ “ “ D ˜ E p q q ` x ω, DG p q q y for all q P W j ` ,r p D, R k q . As D ˜ E p q q “ ´ D ` ´ ∇ ¨ T ` DF p u q ˝ u ´ ˘ ¨ q d µ and x ω, DG p q q y “ ´ ∇ ω ¨ q d µ in the distribution sense, it then follows from µ ą D that ∇ ω P W j ´ ,r p D, R k q thus ω P W j,r p D q Ă C p ¯ D q and (2.3) holds. Changing variables back to Ω we have that (2.2) holds, at least weakly, except itremains to check that ω P B Φ p µ q .We now suppose that ω is not in the subdifferential of B Φ p µ q . Therefore there is ν P M p ¯ D q and a ą ˆ D ω d ν ´ Φ p ν q ě a ` ˆ D ω d µ ´ Φ p µ q “ a ´ Φ p µ q . By regularity of Φ and continuity of ω , we can take ν in W j,r p D q . By convexity of Φ we deduce that forall t P r , s , Φ pp ´ t q µ ` tν q ď ´ at ` Φ p µ q ` t ˆ D ω d ν. By the implicit function theorem and previous linearization argument, we find q t P W j ` ,r p D, R k q forsufficiently small t such that p u ` q t ˝ u q L Ω “ tν ` p ´ t q µ and DG p q q “ ν ´ µ where q is thetemporal derivative of q evaluated at t “
0. We can write E p u ` q t ˝ u q “ ˜ E p q t q ` Φ pp ´ t q µ ` tν q ď ˜ E p q t q ´ at ` Φ p µ q ` t ˆ D ω d ν, and there is equality if t “
0. The derivative of the right hand side at t “ t ˆ ˜ E p q t q ´ at ` Φ p µ q ` t ˆ D ω d ν ˙ˇˇˇˇ t “ “ ´ a ` D ˜ E p q q ` ˆ D ω d ν “ ´ a ´ x ω, DG p q q y W j,q p D q ` ˆ D ω d ν “ ´ a ´ ˆ D ω d p ν ´ µ q ` ˆ D ω d ν “ ´ a. Hence by taking t small enough we see that E p u ` q t ˝ u q ă E p u q , which contradicts the local optimalityof u . (cid:3) Global optimality
Now let us turn to conditions guaranteeing that a solution of (2.2) is a global minimizer of the problem.We start with an easy observation.
Proposition 3.1.
Let u P W ,pg p Ω , D q with Φ p u L Ω q ă `8 and assume that there exists ω P B Φ p u L Ω q such that u is a (unique) global minimizer of (3.1) v ÞÑ ˆ Ω ´ W p ∇ v q ` F p v q ` ω p v q ¯ d L Ω over W ,pg p Ω , D q . Then u is a (unique) minimizer of the energy E introduced in Definition 2.2. HIDDEN CONVEXITY IN A PROBLEM OF NONLINEAR ELASTICITY
Proof.
Indeed, we can write by definition of B Φ p u L Ω q that for any competitor v , E p v q “ ˆ Ω ´ W p ∇ v q ` F p v q ¯ d L Ω ` Φ p v L Ω qě ˆ Ω ´ W p ∇ v q ` F p v q ¯ d L Ω ` ˆ D ω d p v L Ω q ´ Φ ˚ p ω q“ ˆ Ω ´ W p ∇ v q ` F p v q ` ω p v q ¯ d L Ω ´ Φ ˚ p ω qě ˆ Ω ´ W p ∇ u q ` F p u q ` ω p u q ¯ d L Ω ´ Φ ˚ p ω q“ ˆ Ω ´ W p ∇ u q ` F p u q ¯ d L Ω ` Φ p u L Ω q “ E p u q where the second inequality is the assumption on u and the last inequality comes from ω P B Φ p u L Ω q . (cid:3) Then we just notice that (2.2) are also the Euler-Lagrange equations for (3.1). Hence justifying globaloptimality for (3.1) is enough to yield global optimality of our original problem. The first result is thatconvexity of ω implies global optimality of an equilibrium u . Theorem 3.2.
Let u P W ,pg p Ω , D q and assume that there exists ω P B Φ p u L Ω q such that ω can beextended to a convex function on R k and (2.2) holds weakly (where ∇ ω can be any measurable selectionof the subdifferential of ω ). Then u is a global minimizer of the energy E defined in (2.2) . Moreover, if ω , F , or W is strictly convex, u is the unique global minimizer of E .Proof. We just use Proposition 3.1 by noticing that (3.1) is a convex (resp. strictly convex) problemprovided that ω is convex (resp. one of ω , F , or W is strictly convex), and that (2.2) are the EulerLagrange equation for (3.1). Indeed, a solution of the Euler-Lagrange equations of a convex (resp.strictly convex) problem is a global minimizer (resp. the unique global minimizer). (cid:3) Remark . In the case where W p C q “ | C | is quadratic, Φ “ Φ inc while F “ u thelinearized version of our Problem (1.1) are nothing else than the Stokes equations which read $’&’% ∆ u “ ∇ p in Ω ∇ ¨ u “ u “ g on B Ω , and to make the link with our notation we would take p “ ω . As p is harmonic if it is not constant thenit cannot be convex. In particular, Theorem 3.2 will not apply in a situation without exterior forces, atleast for the linearized problem.If we restrict to uniformly convex energies, we can relax the convexity assumption on the pressure. Werecall that a function f : R k Ñ R is λ convex if and only if y ÞÑ f p y q ` λ | y | is a convex function. Theorem 3.4.
Assume that W is λ W convex and F is λ F convex. Let λ p Ω q ą the first eigenvalue ofthe Dirichlet Laplacian on Ω .Let u P W ,pg p Ω , D q for p ě and assume that there exists ω P B Φ p u L Ω q such that ω can be extendedin a λ -convex function on R k with λ ě ´ λ W λ p Ω q ´ λ F and (2.2) holds weakly (where ∇ ω can beany measurable selection of the subdifferential of ω ). Then u is the unique minimizer of the energy E introduced in Definition 2.2. Moreover, if λ ą ´ λ W λ p Ω q ´ λ F , u is the unique minimizer of E .Proof. Thanks to Proposition 3.1, we just need to study the problem (3.1). We know that the function W is λ W -convex. Combining this with the definition of λ p Ω q , it is clear that for any u, w belonging to IDDEN CONVEXITY IN A PROBLEM OF NONLINEAR ELASTICITY 9 W ,pg p Ω , D q (in particular u ´ v vanishes on B Ω) ˆ Ω ´ W p v q ` F p v q ¯ ´ ˆ Ω ´ W p u q ´ F p v q ¯ d L Ω ě ˆ Ω ´ ´ ∇ ¨ p DW p ∇ u qqp v ´ u q ` DF p u q ¨ p v ´ u q ` λ F | v ´ u | ` λ W | ∇ v ´ ∇ u | ¯ d L Ω ě ˆ Ω ´ ´ ∇ ¨ p DW p ∇ u qqp v ´ u q ` DF p u q ¨ p v ´ u q ` λ F ` λ W λ p Ω q | v ´ u | ¯ d L Ω . Hence by adding the functional ´ ω p v q d L Ω we see that the problem (3.1) is p λ F ` λ W λ p Ω q ` λ q convex,which yields global optimality (if λ ě ´ λ W λ p Ω q ´ λ F ) and uniqueness (if λ ą ´ λ W λ p Ω q ´ λ F ) forsolutions of the Euler-Lagrange equations (2.2). (cid:3) As a corollary, we deduce a local optimality result.
Corollary 3.5.
Assume that W is λ W -convex with λ W ą and that Φ is either Φ inc or Φ h . We restrictto the case Ω “ D .Let p u, ω q be a smooth solution of the equilibirum equations (2.2) with u being one to one. Then thereexists r ą such that for every ˜Ω smooth connected subset of Ω of diameter bounded by r , the function u | ˜Ω is the global maximizer of (2.1) with source domain ˜Ω , target domain u p ˜Ω q and boundary conditions u | B ˜Ω .Proof. With the assumption that Φ is either Φ inc or an integral function of the density, one can noticethat ω | ˜Ω P B Φ p u | ˜Ω L ˜Ω q provided that ω P B Φ p u L Ω q .Thus, to apply Theorem 3.4, we just need to notice that λ p ˜Ω q goes to `8 when the diameter of ˜Ωgoes to 0, and that every smooth function is locally λ -convex for some finite λ . (cid:3) Examples
Affine deformations.
Assume that u : Ω Ñ D is an affine mapping and that Φ is either Φ inc oran integral function of the density Φ h , and F “
0. Then it is clear that B Φ p u L Ω q contains a constantfunction ω . As u clearly satisfies ∇ ¨ p DW p ∇ u qq “ p ∇ ω q ˝ u “ , and that ω is convex, we can apply Theorem 3.2.In conclusion, if u is an affine mapping, then it is a global minimizer of the Problem (2.1) with boundaryconditions u | B Ω . Moreover, if W is strictly convex, it is the unique global minimizer.In [18] a stronger result is proved with an additional assumption on Ω: namely that under the as-sumption that Ω is star-shaped and D has the same dimension than Ω, any solution of the equilibriumequations (2.2) with affine boundary conditions is an affine map. Moreover, the authors can allow for W to be any quasiconvex function.4.2. Perturbation of the identity with exterior force.
For simplicity let W be quadratic and werestrict to Φ to be Φ inc the incompressibility constraint. We take Ω “ D and consider the boundaryconditions imposed by g ǫ p x q “ x ` ǫg for g P C ,α p ¯Ω q . We take F p u q “ ´ ∇ ψ ¨ u for ψ P C ,α p ¯Ω q ,which is strictly convex. Then the identity is a global minimizer for ǫ “ ω p y q “ ψ p y q . By the implicit function theorem and analysis of the linearized Stokes equation, we concludethat for some ǫ ą u ǫ P C ,α p Ω , R k q and ω ǫ P C ,α p Ω , R k q for ǫ P r , ǫ q . In particular we can select ǫ ą ω ǫ remains strictly convex and thus Theorem 3.2implies u ǫ remains the unique global minimum. The recent results of [17] show that the local argumentcan be extended globally by means of a topological degree. The solutions remain a global minimum until ω ǫ loses the convexity imparted by ψ . Pure torsion of a cylinder.
In this example let’s take for simplicity W quadratic and (not forsimplicity) we restrict to Φ to be Φ inc the incompressibility constraint. Let’s take Ω “ D “ B p , q ˆr , s Ă R a cylinder. A point x P R will be written x “ p x h , z q P R ˆ R . We will denote by R θ : R Ñ R the rotation by an angle θ .Let a be a parameter. We consider the mapping u a : Ω Ñ Ω defined by u a p x h , z q “ ˆ R az x h z ˙ , that is each horizontal slice is rotated by an angle proportional to z . This mapping always satisfies u L Ω “ L Ω . Moreover, a straightforward computation leads to∆ u a p x h , z q “ ˆ ´ a R az x h ˙ “ p ∇ ω a q ˝ u a p x h , z q provided we define ω a p x h , z q “ ´ a | x h | . In other words, u a satisfies the equilibrium equations (2.2).This is not a surprise, actually u a satisfies the equilibrium equations for much more general W as it isa well undestood result in elasticity theory, see for instance [16, Section 3.3]. We make the followingobservations. • If a is small enough, than u a is the unique global minimizer of (2.1) with boundary conditions u a | B Ω . This is a direct consequence of Theorem 3.4. • If a is large enough, than u a is not a global minimizer of the auxiliary problem (3.1) with boundaryconditions g “ u a | B Ω . Indeed, for any displacement q P W , p Ω , R q with nontrivial horizontalcomponent, q h , the auxiliary energy of u a ` ǫq will decrease for sufficiently large a as ˆ Ω ´ ˇˇ ∇ u a ` ǫ ∇ q ˇˇ ` ω a ` u a ` ǫq ˘¯ d L Ω ´ ˆ Ω ´ ˇˇ ∇ u a ˇˇ ` ω a ` u a ˘¯ d L Ω “ ǫ ˆ Ω ´ ˇˇ ∇ q ˇˇ ´ a ˇˇ q h ˇˇ ¯ d L Ω , where the cancellation of the cross terms occurred as u a is a solution of the equilibrium equation(2.2). For sufficiently small ǫ , if u is bounded then u a ` ǫq P W , g p Ω , D q . • It remains unclear whether there are any other solutions to the equilibrium equations (2.2) andif u a is the global minimum of Problem (3.1) with boundary conditions u a | B Ω for all a .4.4. Multiple local minima.
Our result provides global minimality but does not prevent the existenceof local minimizers or other equilibria. In particular, in [24] the following example is studied.The authors take Ω “ D “ t x P R : 0 ă r ď | x | ď r u Ă R an annulus and as boundaryconditions g they take the identity. The energy Φ is Φ inc enforcing the incompressibility condition. Theidentity map is the unique global minimizer of (2.1). However, if S N Ă W ,pg p Ω , D q is the set of mappings u P W ,pg p Ω , D q such that a.e. radius is mapped by u to a curve of index N , then S N is connected for the W ,p p Ω , D q topology. In particular, the energy E admits a local minimizer on every S N .5. A convex relaxation
The proofs of Theorems 3.2 and 3.4 rely on convexity arguments: the problem (3.1) is convex undersome assumptions on W and the pressure ω . However, the problem (3.1) depends on ω , that is on thesolution. In this section, we present a natural convex relaxation of the original problem which can beformulated in a very general context. Then, under the assumption of λ convexity on ω , we show thatthis convex relaxation is tight.Our result is reminiscent of Brenier’s works (for instance [7, 11]), where he did something similar:taking a non convex problem, formulating a convex relaxation, and finding assumptions on solutions ofthe original problem guaranteeing that they are also solutions of the relaxed problem. IDDEN CONVEXITY IN A PROBLEM OF NONLINEAR ELASTICITY 11
Our relaxation relies on a definition of Dirichlet energy for measure valued mappings proposed byBrenier 20 years ago [9] and investigated more recently by the third author [20].An alternative convex relaxation of the problem has been given in [1] that uses a measure on the spaceof Ω ˆ D ˆ R ` ˆ R dk , where the last two arguments correspond to distributions for det p ∇ u q and ∇ u . Thedual problem they derive results in an unknown k , that relates to the Legendre transform of ω in ourwork. In the convex formulation we present here, the pressure of the deformed configuration, ω , appearsin a manner entirely analogous to the dual Kantorovich potential of optimal transport.5.1. The (primal) convex problem.
The idea is to replace u : Ω Ñ D by a transport plan π P M ` p ¯Ω ˆ ¯ D q whose marginals are L Ω and µ respectively. If we do that, the potential term and thepenalization on the image measure of u are easily translated: what we have gained is that the measure µ “ u L Ω is replaced by µ “ proj D π a linear expression in π . What is less obvious is what to do withthe term involving the gradient of u .To that extent, we extend J P M p ¯Ω ˆ ¯ D, R dk q which is interpreted as the “flux” of the transport plan π . Namely, we will enforce the “generalized continuity equation”(5.1) ∇ Ω π ` ∇ D ¨ J “ , and the stored energy will be replaced by(5.2) ¨ Ω ˆ D W ˆ d J d π ˙ d π which is a jointly convex function of π and J . We explain below in Lemma 5.3 how to embed our originalproblem into this convex relaxation.Specifically, we say that a nonnegative measure π P M ` p ¯Ω ˆ ¯ D q and a matrix valued measure J P M p Ω ˆ D, R dk q satisfy the generalized continuity equation with boundary conditions g if and only if forall ϕ P C p ¯Ω ˆ ¯ D, R d q there holds(5.3) ¨ Ω ˆ D ∇ Ω ¨ ϕ d π ` ¨ Ω ˆ D ∇ D ϕ : d J “ ˆ B Ω ϕ p g q ¨ n Ω d σ, where ϕ p g qp x q “ ϕ p x, g p x qq , n Ω is the outward unit normal and σ is the surface area measure of Ω. Thisis the weak form of (5.1) with the boundary conditions being given by π p x, ¨q “ δ g p x q for x P B Ω. Definition 5.1.
We say that a triple p π, J, µ q where π P M ` p ¯Ω ˆ ¯ D q , J P M p ¯Ω ˆ ¯ D, R dk q and µ P M ` p ¯ D q is admissible if the marginals of π are L Ω and µ respectively and if p π, J q satisfy (5.3) the generalizedcontinuity equation with boundary conditions g .For any admissible triple p π, J, µ q , we define its (relaxed) energy by E r p π, J, µ q “ ¨ Ω ˆ D „ W ˆ d J d π ˙ ` F d π ` Φ p µ q . This energy is convex and the set of admissible triples is a convex set: we have precisely built theseobjects for that purpose.
Remark . As proved in [20], in the case where W p C q “ | C | is quadratic this energy admits a metricformulation.Let us denote by W the quadratic Wasserstein distance on M ` p ¯ D q , extended to `8 if the twomeasures do not have the same total mass (see for instance [25, Chapter 5] for a definition). We fix π P M ` p ¯Ω ˆ ¯ D q whose first marginal is L Ω and we denote by p π x q x P Ω its disintegration w.r.t. the firstcomponent, in particular π x P M ` p ¯ D q for a.e. x P Ω. If D is convex, [20, Theorem 3.26] yieldsmin J ¨ Ω ˆ D ˇˇˇˇ d J d π ˇˇˇˇ d π “ lim ε Ñ C d ¨ Ω ˆ Ω W p π x , π x q ε d ` | x ´ x |ď ε d x d x where the infimum is taken over all J P M p ¯Ω ˆ ¯ D, R dk q such that p π, J q satisfy (5.3) and C d is adimensional constant.The right hand side in the equation above can be interpreted as a metric definition of the Dirichletenergy [19] for the mapping x ÞÑ π x valued in M ` p ¯ D q endowed with the distance W .The relaxed energy is a convex relaxation of our original energy in the following sense. Lemma 5.3.
Let u P W ,pg p Ω , D q be given. Let us define µ u “ u L Ω while π u and J u are defined by, forany test functions a P C p ¯Ω ˆ ¯ D q and B P C p ¯Ω ˆ ¯ D, R dk q , ¨ Ω ˆ D a d π u “ ˆ Ω a p x, u p x qq d L Ω p x q and ¨ Ω ˆ D B d J u “ ˆ Ω B p x, u p x qq : ∇ u p x q d L Ω p x q . Then p π u , J u , µ q is admissible and E r p π u , J u , µ u q “ E p u q . We leave the proof as an exercise to the reader, see [20, Proposition 5.2] where it is written explicitly.Note that it could be written π u “ δ u L Ω and J u “ ∇ u δ u L Ω .5.2. The dual problem.
Let us write the dual of the problem above. It can be guessed by a formalinf ´ sup exchange analogous to what was done in [9] and [20]. The absence of duality gap could beobtained via Fenchel-Rockafellar theorem as written in [20], however we will not need it hence we willnot prove it. Dual attainment is an open question.There will be three dual variables: a Lagrange multiplier ϕ for the generalized continuity equation,and then two Lagrange multipliers ψ, ω for the marginal constraints on π . Definition 5.4.
We say that a triple p ϕ, ψ, ω q where ϕ P C p ¯Ω ˆ ¯ D, R d q , ψ P C p ¯Ω q and ω P C p ¯ D q isadmissible if for all x, y P Ω ˆ D ,(5.4) ψ p x q ` ω p y q ` F p x, y q ě ∇ Ω ¨ ϕ p x, y q ` W ˚ ` x, ∇ D ϕ p x, y q ˘ . For any admissible triple p ϕ, ψ, ω q , we define its (relaxed) dual energy by E ˚ r p ϕ, ψ, ω q “ ˆ B Ω ϕ p g q ¨ n Ω d σ ´ ˆ Ω ψ d L Ω ´ Φ ˚ p ω q . Remark . In the case where there is no integral energy, that is we take ϕ “
0, and if Φ “ Φ inc , thenour dual problem is exactly Kantorvich dual problem [25, Chapter 1] for the cost ´ F . Proposition 5.6 (Weak duality) . Let p π, J, µ q be admissible for the primal problem and p ϕ, ψ, ω q admis-sible for the dual problem. Then E r p π, J, µ q ě E ˚ r p ϕ, ψ, ω q and equality holds if and only if $’’&’’% ∇ D ϕ p x, y q “ DW ˆ x, d J d π p x, y q ˙ for π -a.e. p x, y q P Ω ˆ D, equality holds in (5.4) for π -a.e. p x, y q P Ω ˆ D,ω
P B Φ p µ q . As always in optimal transport, we perpetuate the confusion as the dual should be rather called primal: measures arethe dual of continuous functions and not the other way around.
IDDEN CONVEXITY IN A PROBLEM OF NONLINEAR ELASTICITY 13
Proof.
Let’s compute the difference: we take p π, J, µ q admissible for the primal problem and p ϕ, ψ, ω q admissible for the dual problem. Then E r p π, J, µ q ´ E ˚ r p ϕ, ψ, ω q“ ¨ Ω ˆ D „ W ˆ d J d π ˙ ` F d π ` Φ p µ q ´ ˆ B Ω ϕ p g q ¨ n Ω d σ ` ˆ Ω ψ d L Ω ` Φ ˚ p ω qě ¨ Ω ˆ D „ W ˆ d J d π ˙ ` F d π ` ˆ Ω ψ d L Ω ` ˆ D ω d µ ´ ˆ B Ω ϕ p g q ¨ n Ω d σ “ ¨ Ω ˆ D „ W ˆ d J d π ˙ ` F ` ψ ` ω d π ´ ˆ B Ω ϕ p g q ¨ n Ω d σ, where we have used the definition of Φ ˚ and then the assumption that the marginals of π are L Ω and µ .Using the generalized continuity equation and integrating the constraint (5.4) w.r.t. π , ¨ Ω ˆ D „ W ˆ d J d π ˙ ` F ` ψ ` ω d π ´ ˆ B Ω ϕ p g q ¨ n Ω d σ “ ¨ Ω ˆ D „ W ˆ d J d π ˙ ` F ` ψ ` ω ´ ∇ Ω ¨ ϕ ´ ∇ D ϕ : DW ˆ d J d π ˙ d π ě ¨ Ω ˆ D „ W ˆ d J d π ˙ ` W ˚ ` ∇ D ϕ ˘ ´ ∇ D ϕ : DW ˆ d J d π ˙ d π. Eventually, using the definition of W ˚ we get E r p π, J, µ q ´ E ˚ r p ϕ, ψ, ω q ě . Tracking back all the inequalities, we deduce the necessary and sufficient conditions for the absence ofduality gap. (cid:3)
A particular solution for the dual.
So far, putting together Lemma 5.3 and Proposition 5.6, wehave sup p ϕ,ψ,ω q admissible E ˚ r p ϕ, ψ, ω q ď min p π,J,µ q admissible E r p π, J, µ q ď min u P W ,pg p Ω ,D q E p u q . To prove that everything boils down to a big equality, it is sufficient to provide one competitor in thedual which matches E p u q .We handle this for the case of the Dirichlet energy, for which we can extend the result to λ convexpressure as in Theorem 3.4. We recall that λ p Ω q ą Proposition 5.7.
Let W p C q “ | C | and assume F is smooth and λ F convex in its second variable.We suppose that u is a smooth solution of the Euler-Lagrange equation (2.2) for some ω P B Φ p u L Ω q .Assume ω is λ -convex with λ ą ´ λ p Ω q ´ λ F . Then there exists p ϕ u , ψ u , ω u q admissible such that E ˚ r p ϕ u , ψ u , ω u q “ E p u q . Proof.
Recall that if we define p π u , J u , µ u q as in Lemma 5.3 we just need to prove that E ˚ r p ϕ u , ψ u , ω u q “ E r p π u , J u , µ u q We take ω u “ ω . In particular ω u P B Φ p µ u q .We choose ε ą λ ` λ p Ω q` λ F ą ε . Let w P C p ¯Ω , R d q be a function such that ∇ ¨ w `| w | ă´ λ p Ω q ` ε : see Lemma 5.9 below for the existence of such an w . We define ϕ u , which is valued in R d and that we see as a row vector as ϕ u p x, y q “ y J ∇ u p x q ` | u p x q ´ y | w p x q J . Without the quadratric term (that is if w “
0) we retrieve the solution proposed by Brenier in [9] for thecase where there is no constraint on µ , that is Φ “
0. This quadratic term will be crucial to go from ω convex to ω that is λ -convex. Taking the derivative of ϕ u w.r.t. y and evaluating at y “ u p x q , we get ∇ D ϕ u p x, u p x qq “ ∇ u p x q which exactly reads as ∇ D ϕ u p x, y q “ DW ˆ d J u d π u p x, y q ˙ for π u -a.e. p x, y q P Ω ˆ D. It remains to choose the function ψ u such that the constraint (5.4) is satisfied. Let us compute theright hand side of (5.4): we find ∇ Ω ¨ ϕ u p x, y q ` | ∇ D ϕ u p x, y q| (5.5) “ y J ∆ u p x q ` p ∇ ¨ w qp x q | y ´ u p x q| ´ p u p x q ´ y q J ∇ u p x q w p x q ` ˇˇ ∇ u p x q ` w p x qp u p x q ´ y q J ˇˇ “ y J ∆ u p x q ` | ∇ u p x q| ` ` ∇ ¨ w p x q ` | w p x q| ˘ | y ´ u p x q| ď y J ∆ u p x q ` | ∇ u p x q| ` ´ λ p Ω q ` ε | y ´ u p x q| , where the last inequality derives from the way w was chosen and is an equality if y “ u p x q . Notice thatsome cancellation of the cross term occurred because of the quadratic structure of the problem. Giventhis computation, we define ψ u p x q “ ´ ω u ` u p x q ˘ ´ F p x, u p x qq ` p ∇ ω u q ` u p x q ˘ ¨ u p x q ` DF p x, u p x qq ¨ u p x q ` | ∇ u p x q| . To see if (5.4) is satisfied we compute ψ u p x q ` ω u p y q ` F p x, y q ´ ∇ Ω ¨ ϕ u p x, y q ´ | ∇ D ϕ u p x, y q| ě ω u p y q ` F p x, y q ´ ω u p u p x qq ´ F p x, u p x qq ` p ∇ ω u q ` u p x q ˘ ¨ u p x q ` DF p x, u p x qq ¨ u p x q´ y J ∆ u p x q ` λ p Ω q ´ ε | y ´ u p x q| “ ω u p y q ` F p x, y q ´ ω u ` u p x q ˘ ´ F p x, u p x qq ´ “ p ∇ ω u q ` u p x q ˘ ` DF p x, u p x qq ‰ ¨ ` y ´ u p x q ˘ ` λ p Ω q ´ ε | y ´ u p x q| ě λ ` λ F ` λ p Ω q ´ ε | y ´ u p x q| . In this computation, the equality derives from the equilibrium equations (2.2) and the last inequalitycomes from the p λ ` λ F q -convexity of y ÞÑ ω p y q ` F p x, y q . But ε was chosen in such a way that λ ` λ F ` λ p Ω q ´ ε ą p ϕ u , ψ u , ω u q satisfies (5.4) and there is equality if and only if y “ u p x q ,that is on the support of π u . (cid:3) Remark . The cancellation that occurred in (5.5) is crucial to get the results and is what wouldbreak down if we replaced the Dirichlet energy by ´ Ω W p ∇ u q d L Ω with W convex. However, in thecase that ω is convex with general energy, we obtain the same result as Proposition 5.7 by choosing ϕ u p x, y q “ y J DW ` ∇ u p x q ˘ .During the proof we have used the following lemma, which relies on a standard change of variables innonlinear elliptic equations. IDDEN CONVEXITY IN A PROBLEM OF NONLINEAR ELASTICITY 15
Lemma 5.9.
Let λ ą ´ λ p Ω q . Then there exists w P C p ¯Ω , R d q such that ∇ ¨ w ` | w | ă λ everywhere on Ω .Proof. We will look rather for z P C p ¯Ω q such that∆ z ` | ∇ z | ă λ as we can always take w “ ∇ z . To that extent, let f a smooth strictly positive function such that∆ f ď λf . This is always possible by mollifying the first eigenvalue of the Dirichlet Laplacian on Ω. Thendefining z “ ln f works as we leave the reader to check. (cid:3) Another proof of Theorem 3.4.
We can give another proof of Theorem 3.4 for the Dirichletenergy, that is, when W p C q “ C , relying on this convex relaxation. Proof of Theorem 3.4.
Let u a smooth solution of the Euler-Lagrange equation (2.2) for some ω PB Φ p u L Ω q . We assume ω is λ -convex with λ ą ´ λ p Ω q ´ λ F .Let p ϕ u , ψ u , ω u q be the optimal solution of the dual problem built in the proof of Proposition 5.7. Forany competitor v P W , g p Ω , D q there holds E p v q ě E r p π v , J v , µ v q ě E ˚ r p ϕ u , ψ u , ω u q “ E p u q where we have used successively Lemma 5.3, Proposition 5.6 and then Proposition 5.7. Moreover, if thereis equality then by Proposition 5.6 the constraint (5.4) (with p ϕ u , ψ u , ω u q ) must be an equality on thesupport of π v , that is for y “ v p x q . However we have seen in the proof of Proposition 5.6 that equalityhappens only for y “ u p x q . This yields u “ v . (cid:3) Acknowledgments
The authors are partially supported by the Natural Sciences and Engineering Research Council ofCanada (NSERC) through the Discovery Grant program. HL is partially supported by the PacificInstitute for the Mathematical Sciences (PIMS) through a PIMS postdoctoral fellowship.
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