Jean-Philippe Mandallena

Multi-parameter homogenization by localization and blow-up

We give an alternative self-contained proof of the homogenization theorem for periodic multi-parameter integrals that was established by the authors. The proof in that paper relies on the so-called compactness method for

On the relaxation of variational integrals in metric Sobolev spaces

\Gamma

On the regularity of solutions of one-dimensional variational obstacle problems

-convergence, while the one presented here is by direct verification: the candidate to be the limit homogenized functional is first exhibited and the definition of

{\Gamma}-convergence of nonconvex integrals in Cheeger-Sobolev spaces and homogenization

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Relaxation theorems in nonlinear elasticity

-convergence is then verified. This is done by an extension of bounded gradient sequences using the Acerbi et al. extension theorem from connected sets, and by the adaptation of some localization and blow-up techniques developed by Fonseca and Muller, together with De Giorgis slicing method.

Interchange of infimum and integral

Abstract 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–Keiths differentiable structure.

Relaxation of variational problems in two-dimensional nonlinear elasticity

Abstract We study the regularity of solutions of one-dimensional variational obstacle problems in W 1 , 1 {W^{1,1}} when the Lagrangian is locally Hölder continuous and globally elliptic. In the spirit of the work of Sychev [5, 6, 7], a direct method is presented for investigating such regularity problems with obstacles. This consists of introducing a general subclass ℒ {\mathcal{L}} of W 1 , 1 {W^{1,1}} , related in a certain way to one-dimensional variational obstacle problems, such that every function of ℒ {\mathcal{L}} has Tonelli’s partial regularity, and then to prove that, depending on the regularity of the obstacles, solutions of corresponding variational problems belong to ℒ {\mathcal{L}} . As an application of this direct method, we prove that if the obstacles are C 1 , σ {C^{1,\sigma}} , then every Sobolev solution has Tonelli’s partial regularity.

The nonlinear membrane energy: variational derivation under the constraint “det∇u>0”

Abstract We study Γ-convergence of nonconvex variational integrals of the calculus of variations in the setting of Cheeger–Sobolev spaces. Applications to relaxation and homogenization are given.