Omar Anza Hafsa

On the relaxation of variational integrals in metric Sobolev spaces

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.

On a homogenization technique for singular integrals

We study homogenization by Γ -convergence of functionals of type � Ω W � x e , ∇φ(x) � dx, where Ω ⊂ R N is a bounded open set, φ ∈ W 1,p (Ω; R m )a nd p> 1, when the 1-periodic integrand W : R N × M m×N → (0, +∞) is not of p-polynomial growth. Our homogenization technique can be applied when m = N and W has a singular behavior of type W (x, ξ) → +∞ as det ξ → 0. However, our technique is not consistent with the constraint W (x, ξ) =+ ∞ if and only if det ξ 0.

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

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.