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Jean-Philippe Mandallena

Publications and source records attributed to Jean-Philippe Mandallena.

18 recordsLinked to original sources
arXiv2016

On the regularity of solutions of one dimensional variational obstacle problems

We study the regularity of solutions of one dimensional variational obstacle problems in $W^{1,1}$ when the Lagrangian is locally Hölder continuous and globally elliptic. In the spirit of the work of Sychev ([Syc89, Syc91, Syc92]), a direct method is presented for investigating such regularity problems with obstacles. This consists of introducing a general subclass $\mathcal{L}$ of $W^{1,1}$, related in a certain way to one dimensional variational obstacle problems, such that every function of $\mathcal{L}$ has Tonelli's partial regularity, and then to prove that, depending of the regularity of the obstacles, solutions of corresponding variational problems belong to $\mathcal{L}$. As an application of this direct method, we prove that if the obstacles are $C^{1,σ}$ then every Sobolev solution has Tonelli's partial regularity.

math.CA↗
arXiv2015

$Γ$-limits of functionals determined by their infima

We study the integral representation of $Γ$-limits of $p$-coercive integral functionals of the calculus of variations in the spirit of \cite{dalmaso-modica86}. We use infima of local Dirichlet problems to characterize the limit integrands. Applications to homogenization and relaxation are given.

math.CA↗
arXiv2013

On the relaxation of variational integrals in metric Sobolev spaces

We give an extension of the theory of relaxation of variational integrals in classical Sobolev spaces to the setting of metric Sobolev spaces. More precisely, we establish a general framework to deal with the problem of finding an integral representation for relaxed variational functionals of variational integrals of the calculus of variations in the setting of metric measure spaces. We prove integral representation theorems, both in the convex and non-convex case, which extend and complete previous results in the setting of euclidean measure spaces to the setting of metric measure spaces. We also show that these integral representation theorems can be applied in the setting of Cheeger-Keith's differentiable structure.

math.CA↗
arXiv2013

Radial representation of lower semicontinuous envelope

We give an extension to a nonconvex setting of the classical radial representation result for lower semicontinuous envelope of a convex function on the boundary of its effective domain. We introduce the concept of radial uniform upper semicontinuity which plays the role of convexity, and allows to prove a radial representation result for nonconvex functions. An application to the relaxation of multiple integrals with constraints on the gradient is given.

math.CA↗
arXiv2013

Localization principle and relaxation

Relaxation theorems for multiple integrals on W^{1,p}(Ω;\RR^m), where p\in]1,\infty[, are proved under general conditions on the integrand L:\MM\to[0,\infty] which is Borel measurable and not necessarily finite. We involve a localization principle that we previously used to prove a general lower semicontinuity result. We apply these general results to the relaxation of nonconvex integrals with exponential-growth.

math.CA↗
arXiv2012

Lower semicontinuity via W^{1,q}-quasiconvexity

We isolate a general condition, that we call "localization principle", on the integrand L:\MM\to[0,\infty], assumed to be continuous, under which W^{1,q}-quasiconvexity with q\in[1,\infty] is a sufficient condition for I(u)=\int_ΩL(\nabla u(x))dx to be sequentially weakly lower semicontinuous on W^{1,p}(Ω;\RR^m) with p\in]1,\infty[. Some applications are given.

math.CA↗
arXiv2011

Relaxation and 3d-2d passage theorems in hyperelasticity

We give an overview of relaxation and 3d-2d passage theorems in hyperelasticity in the framework of the multidimensional calculus of variations. Some open questions are addressed. This paper, which is an expanded version of the outline-paper [AHM09b], comes as a companion to [AHM09a].

math.AP↗
arXiv2010

Homogenization of singular integrals in W^{1,\infty}

A periodic homogenization result of nonconvex integral functionals in the vectorial case with convex bounded constraints on gradients is proved. The class of integrands considered have singular behavior near the boundary of the convex set of the constraints. We apply the result to the case of periodic homogenization in hyperelasticity for bounded gradients of deformations.

math.AP↗
arXiv2009

Relaxation et passage 3D-2D avec contraintes de type déterminant

The goal of this paper is, on the one hand, to make available a set of tools and methods to deal with relaxation and 3D-2D passage with determinant type constraints, and, on the other hand, to give a complete proof of the 3D-2D passage by Gamma-convergence under the constraint determinant positive.

math.AP↗
arXiv2006

Relaxation theorems in nonlinear elasticity

Relaxation theorems which apply to one, two and three-dimensional nonlinear elasticity are proved. We take into account the fact an infinite amount of energy is required to compress a finite line, surface or volume into zero line, surface or volume. However, we do not prevent orientation reversal.

math.CA↗