We study the regularity of solutions of one dimensional variational obstacle problems in $W^{1,1}$ when the Lagrangian is locally H\"older 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 $C^{1,\sigma}$ then every Sobolev solution has Tonelli's partial regularity.

Jean-Philippe Mandallena

We study homogenization by Gamma-convergence of periodic multiple integrals of the calculus of variations when the integrand can take infinite values outside of a convex set of matrices.

Omar Anza Hafsa Jean-Philippe Mandallena

We study homogenization by $\Gamma$-convergence of periodic nonconvex integrals when the integrand has quasiconvex growth with convex effective domain.

Omar Anza Hafsa Jean-Philippe Mandallena Hamdi Zorgati

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.

Omar Anza Hafsa Jean-Philippe Mandallena

We study the integral representation of $\Gamma$-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.

Omar Anza Hafsa Jean-Philippe Mandallena

Acerbi, Buttazzo and Percivale gave a variational definition of the nonlinear string energy under the constraint "det$\nabla u>0$" (see [1]). In the same spirit, we obtain the nonlinear membrane energy under the simpler constraint "det$\nabla u\not=0$".

Omar Anza Hafsa Jean-Philippe Mandallena

We study the relaxation of multiple integrals of the calculus of variations, where the integrands are nonconvex with convex effective domain and can take the value \infty. We use local techniques based on measure arguments to prove integral representation in Sobolev spaces of functions which are almost everywhere differentiable. Applications are given in the scalar case and in the case of integrands with quasiconvex growth and p(x)-growth.

Omar Anza Hafsa Jean Philippe Mandallena

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.

Omar Anza Hafsa Jean-Philippe Mandallena

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.

Omar Anza Hafsa Jean-Philippe Mandallena

We establish for smooth projective real curves the equivalent of the classical Clifford inequality known for complex curves. We also study the cases when equality holds.

Jean-Philippe Monnier