Guy Bouchitté - Academia.edu (original) (raw)

Papers by Guy Bouchitté

$Research paper thumbnail of A new <math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><msup><mi>L</mi><mi mathvariant="normal">∞</mi></msup></mrow><annotation encoding="application/x-tex">L^\infty</annotation></semantics></math>L∞ estimate in optimal mass transport$

Proceedings of the American Mathematical Society, 2007

Let Ω be a bounded Lipschitz regular open subset of R d and let µ, ν be two probablity measures o... more Let Ω be a bounded Lipschitz regular open subset of R d and let µ, ν be two probablity measures on Ω. It is well known that if µ = f dx is absolutely continuous, then there exists, for every p > 1, a unique transport map T p pushing forward µ on ν and which realizes the Monge-Kantorovich distance W p (µ, ν). In this paper, we establish an L ∞ bound for the displacement map T p x − x which depends only on p, on the shape of Ω and on the essential infimum of the density f .

Comptes Rendus Mathematique, 2002

On considère le problème de positionnement optimal de n centres de production pour une répartitio... more On considère le problème de positionnement optimal de n centres de production pour une répartition non uniforme de la population. Le critère d'optimisation est une moyenne pondérée de la fonction distance au centre de production le plus proche. Dans cette Note, on étudie le comportement asymptotique du problème quand n tend vers l'infini en le reliant à l'asymptotique d'un problème de transport de masse de type Monge-Kantorovich. Pour citer cet article : G.

Proceedings of the Royal Society A: Mathematical, Physical and Engineering Sciences, 2019

In many applications of structural engineering, the following question arises: given a set of for... more In many applications of structural engineering, the following question arises: given a set of forces f 1 , f 2 , …, f N applied at prescribed points x 1 , x 2 , …, x N , under what constraints on the forces does there exist a truss structure (or wire web) with all elements under tension that supports these forces? Here we provide answer to such a question for any configuration of the terminal points x 1 , x 2 , …, x N in the two- and three-dimensional cases. Specifically, the existence of a web is guaranteed by a necessary and sufficient condition on the loading which corresponds to a finite dimensional linear programming problem. In two dimensions, we show that any such web can be replaced by one in which there are at most P elementary loops, where elementary means that the loop cannot be subdivided into subloops, and where P is the number of forces f 1 , f 2 , …, f N applied at points strictly within the convex hull of x 1 , x 2 , …, x N . In three dimensions, we show th...

ESAIM: Control, Optimisation and Calculus of Variations, 2017

We consider optimization problems for cost functionals which depend on the negative spectrum of S... more We consider optimization problems for cost functionals which depend on the negative spectrum of Schrödinger operators of the form −Δ + V (x), where V is a potential, with prescribed compact support, which has to be determined. Under suitable assumptions the existence of an optimal potential is shown. This can be applied to interesting cases such as costs functions involving finitely many negative eigenvalues.

ESAIM: Control, Optimisation and Calculus of Variations, 2014

], a theory for artificial magnetism in two-dimensional photonic crystals has been developed for ... more ], a theory for artificial magnetism in two-dimensional photonic crystals has been developed for large wavelength using homogenization techniques. In this paper we pursue this approach within a rigorous stochastic framework: dielectric parallel nanorods are randomly disposed, each of them having, up to a large scaling factor, a random permittivity ε(ω) whose law is represented by a density on a window Δ h = [a − , a + ] × [0, h] of the complex plane. We give precise conditions on the initial probability law (permittivity, radius and position of the rods) under which the homogenization process can be performed leading to a deterministic dispersion law for the effective permeability with possibly negative real part. Subsequently a limit analysis h → 0, accounting a density law of ε which concentrates on the real axis, reveals singular behavior due to the presence of resonances in the microstructure.

Zeitschrift für angewandte Mathematik und Physik, 2007

Damage of a linearly-responding material that can completely disintegrate is addressed at small s... more Damage of a linearly-responding material that can completely disintegrate is addressed at small strains. Using time-varying Dirichlet boundary conditions we set up a rateindependent evolution problem in multidimensional situations. The stored energy involves the gradient of the damage variable. This variable as well as the stress and energies are shown to be well defined even under complete damage, in contrast to displacement and strain. Existence of an energetic solution is proved, in particular, by detailed investigating the Γ-limit of the stored energy and its dependence on boundary conditions. Eventually, the theory is illustrated on a one-dimensional example.

SIAM Journal on Mathematical Analysis, 2001

ABSTRACT To the aim of studying the homogenization of low-dimensional periodic structures, we ide... more ABSTRACT To the aim of studying the homogenization of low-dimensional periodic structures, we identify each of them with a periodic positive measure μ on double-struck R signn. We introduce a new notion of two-scale convergence for a sequence of functions vε ∈ Lpμε(Ω;double-struck R signd), where Ω is an open bounded subset of double-struck R signn, and the measures με are the ε-scalings of μ, namely, με(B) := εn μ(ε-1 B). Enforcing the concept of tangential calculus with respect to measures and related periodic Sobolev spaces, we prove a structure theorem for all the possible two-scale limits reached by the sequences (uε, ▽uε) when {uε} ⊂ C10(Ω) satisfy the boundedness condition supε (latin small letter esh)Ω |uε|p + |▽uε|p dμε < +∞ and when the measure μ satisfies suitable connectedness properties. This leads us to deduce the homogenized density of a sequence of energies of the form (latin small letter esh)Ω j(x/ε, ▽u) dμε, where j(y, z) is a convex integrand, periodic in y, and satisfying a p-growth condition. The case of two parameter integrals is also investigated, in particular for what concerns the commutativity of the limit process.

Proceedings of the Royal Society A: Mathematical, Physical and Engineering Sciences, 2009

The classical energy minimization principles of Dirichlet and Thompson are extended as minimizati... more The classical energy minimization principles of Dirichlet and Thompson are extended as minimization principles to acoustics, elastodynamics and electromagnetism in lossy inhomogeneous bodies at a fixed frequency. This is done by building upon the ideas of Cherkaev and Gibiansky, who derived minimization variational principles for quasistatics. In the absence of free current, the primary electromagnetic minimization variational principles have a minimum, which is the time-averaged electrical power dissipated in the body. The variational principles provide constraints on the boundary values of the fields when the moduli are known. Conversely, when the boundary values of the fields have been measured, then they provide information about the values of the moduli within the body. This should have applications to electromagnetic tomography. We also derive saddle-point variational principles that correspond to the variational principles of Gurtin, Willis and Borcea.

Proceedings of the Royal Society of Edinburgh: Section A Mathematics, 1998

The characterisation of the bulk energy density of the relaxation inW1, P(Ω; ℝd) of a functionali... more The characterisation of the bulk energy density of the relaxation inW1, P(Ω; ℝd) of a functionalis obtained forp>q–q/N, whereu∈W1,P(Ω; ℝd), andfis a continuous function on the set ofd × Nmatrices verifyingfor some constantC> 0 and 1 ≦q< + ∞. Typical examples may be found in cavitation and related theories. Standard techniques cannot be used due to the gap between the exponentqof the growth condition and the exponent p of the integrability of the macroscopic strain ∇u. A recently introduced global method for relaxation and fine Sobolev trace and extension theorems are applied.

Nonlinear Analysis: Theory, Methods & Applications, 1990

Journal of the European Mathematical Society, 2001

We study some problems of optimal distribution of masses, and we show that they can be characteri... more We study some problems of optimal distribution of masses, and we show that they can be characterized by a suitable Monge-Kantorovich equation. In the case of scalar state functions, we show the equivalence with a mass transport problem, emphasizing its geometrical approach through geodesics. The case of elasticity, where the state function is vector valued, is also considered. In both cases some examples are presented.

Journal of Functional Analysis, 2003

For a given positive measure m on R n ; we consider integral functionals of the kind F ðuÞ ¼ Z R ... more For a given positive measure m on R n ; we consider integral functionals of the kind F ðuÞ ¼ Z R n f ðx; ru; r 2 uÞ dm; uAC N 0 ðR n Þ; and we study their relaxation with respect to the L p m topology, p being the growth exponent of f : To obtain the relaxed energy F ; we develop a suitable theory of second-order m-intrinsic operators, related to a Cosserat vector field and to a curvature tensor. Our main theorem shows that the functional F is in general a non-local one; this unexpected feature occurs even in very simple examples, when m is the one-dimensional Hausdorff measure over a closed Lipschitz curve in the plane.

Journal of Functional Analysis, 1988

In duality pairs such as (db, WOO) and (W-',p', W$p). a convex integral functional on the space o... more In duality pairs such as (db, WOO) and (W-',p', W$p). a convex integral functional on the space of functions has a polar which admits an integral representation. This representation is the sum of a first term involving the absolutely continuous component of the measure and of a second one which is a positively homogeneous function of the singular part. The duality is useful in plasticity theory. In the Sobolev case the study of non-parametric integrands is new. A description of the sub-differential is obtained.

Journal de Mathématiques Pures et Appliquées, 2011

Given a density function f on a compact subset of R d we look at the problem of finding the best ... more Given a density function f on a compact subset of R d we look at the problem of finding the best approximation of f by discrete measures ν = c i δ x i in the sense of the p-Wasserstein distance, subject to size constraints of the form h(c i) α where h is a given weight function. This is an important problem with applications in economic planning of locations and in information theory. The efficiency of the approximation can be measured by studying the rate at which the minimal distance tends to zero as α tends to infinity. In this paper, we introduce a rescaled distance which depends on a small parameter and establish a representation formula for its limit as a function of the local statistics for the distribution of the c i 's. This allows to treat the asymptotic problem as α → ∞ for a quite large class of weight functions h.

Journal de Mathématiques Pures et Appliquées, 2002

For a given amount m of mass, we study the class of materials which can be reached by homogenizat... more For a given amount m of mass, we study the class of materials which can be reached by homogenization distributing the mass m on periodic structures of prescribed dimension k n in R n. Both in the scalar case of conductivity and in the vectorvalued case of elasticity, we find some bounds for the effective coefficients, depending on the mass m and the dimension parameters k, n. In the scalar case we prove that such bounds are optimal, as they do describe the set of all materials reachable by homogenization of structures of the type under consideration; in the vector-valued case we show that some of our estimates are attained.  2002 Éditions scientifiques et médicales Elsevier SAS. All rights reserved. Résumé Nous étudions la classe des matériaux qui peuvent être obtenus par homogénéisation en répartissant une masse fixée m sur des structures périodiques de dimension k n dans R n. Des bornes sur le tenseur homogénéisé sont obtenues dans le cas de l'équation de la conduction ainsi que dans le cas de l'élasticité linéaire. Dans le premier cas, ces bornes s'avèrent optimales du fait qu'elles caractérisent entièrement l'ensemble des matrices effectives. Dans le cas de l'elasticité nous vérifions que certaines de nos bornes sont atteintes.  2002 Éditions scientifiques et médicales Elsevier SAS. All rights reserved.

Discrete & Continuous Dynamical Systems - A, 2005

In the homogenization of second order elliptic equations with periodic coefficients, it is well k... more In the homogenization of second order elliptic equations with periodic coefficients, it is well known that the rate of convergence of the corrector u n − u hom in the L 2 norm is 1/n, the same as the scale of periodicity (see Jikov et al [7]). It is possible to have the same rate of convergence in the case of almost periodic coefficients under some stringent structural conditions on the coefficients (see Kozlov [8]). The goal of this note is to construct almost periodic media where the rate of convergence is lower than 1/n. To that aim, in the one dimensional setting, we introduce a family of random almost periodic coefficients for which we compute, using Fourier series analysis, the mean rate of convergence r n (mean with respect to the random parameter). This allows us to present examples where we find r n 1/n r for every r > 0, showing a big contrast with the random case considered by Bourgeat and Piatnitski [3] where r n ∼ 1/ √ n.

Comptes Rendus Mathematique, 2004

The homogenization of periodic dielectric structures in harmonic regime usually leads to an effec... more The homogenization of periodic dielectric structures in harmonic regime usually leads to an effective permittivity tensor ε eff. It has been observed by Bouchitté and Felbacq [Waves Random Media 7 (1997) 245-256], that in the high contrast case (high conductivity fibers), this tensor depends on the angular frequency ω. In this Note, we enlight a new effect induced by microscopic resonances which leads in parallel to a possibly negative effective permeability µ eff (ω) (although the original medium is assumed to be nonmagnetic i.e. µ = 1). To cite this article: G.

Comptes Rendus Mathematique, 2010

We consider the variational problem which consists in minimizing the compliance of a prescribed a... more We consider the variational problem which consists in minimizing the compliance of a prescribed amount of isotropic elastic material placed into a given design region when it is subjected to a given load. We perform the asymptotics of this problem when the design region is a straight cylinder with infinitesimal cross section. The results presented in this note concern the pure torsion regime and state the existence of optimal shapes for the limit problem. When the filling ratio tends in turn to zero, these optimal shapes concentrate on the boundary of the Cheeger set of the section of the design region. Résumé Optimisation de la structure d'une poutre fine en torsion et ensembles de Cheeger. On considère le problème d'optimisation suivant : une quantité fixée d'un matériauélastique isotrope donné doitêtre placée dans un cylindre droit de manièreà maximiser sa résistanceà un chargement donné tendantà provoquer un mouvement de torsion. Lorsque le rayon et le taux de remplissage du cylindre tendent tous deux vers zéro, on montre que la distribution optimale de matière se concentre dans chaque section sur le bord de l'ensemble de Cheeger. Version française abrégée Nous considérons une suite de problèmes d'optimisation de forme où la région de design est un cylindre de la forme Q δ = δ D × I, où δ > 0 est un petit paramètre, D est un ouvert borné et connexe de R 2 et I = [−1/2, 1/2]. Un matériauélastique linéaire isotrope (caractérisé par son potentielélastique de déformation j(z) = (λ/2)(tr(z)) 2 + η |z| 2) doitêtre réparti de facon optimale dans un domaine Ω ⊂ Q δ

Comptes Rendus de l'Académie des Sciences - Series I - Mathematics, 1998

We study the homogenization of degenerate elliptic equations like: {−div(aε∇u)=fonΩ,u=u0onΓ0⊂∂Ω,(... more We study the homogenization of degenerate elliptic equations like: {−div(aε∇u)=fonΩ,u=u0onΓ0⊂∂Ω,(1)∂u∂n=0on∂Ω\Γ0, where the coefficent aɛ is ɛ-periodic, takes values of order 1 on a subset Tɛ⊂ Ω (fiber structure) and of order ɛ2 on Ωɛ. We obtain a non-local effective law which cannot be derived from the theory of bounds (see [8]) by assuming that aɛ is lower bounded by a

Calculus of Variations and Partial Differential Equations, 2003

In this paper we present a minimality criterion for the Mumford-Shah functional, and more general... more In this paper we present a minimality criterion for the Mumford-Shah functional, and more generally for non convex variational integrals on SBV which couple a surface and a bulk term. This method provides short and easy proofs for several minimality results.