Homogenization of variational problems in manifold valued Sobolev spaces

TL;DR

This paper investigates the homogenization of variational problems constrained to manifold-valued Sobolev spaces, introducing tangential homogenization and establishing $ ext{Gamma}$-convergence results for energies with various growth conditions.

Contribution

It defines tangential homogenization for manifold-constrained energies and proves $ ext{Gamma}$-convergence results in Sobolev spaces, extending homogenization theory to manifold-valued functions.

Findings

Established $ ext{Gamma}$-convergence for superlinear and linear growth energies.

Introduced the notion of tangential homogenization analogous to tangential quasiconvexity.

Extended homogenization results to manifold-valued Sobolev spaces.

Abstract

Homogenization of integral functionals is studied under the constraint that admissible maps have to take their values into a given smooth manifold. The notion of tangential homogenization is defined by analogy with the tangential quasiconvexity introduced by Dacorogna, Fonseca, Maly and Trivisa \cite{DFMT}. For energies with superlinear or linear growth, a $\Gamma$-convergence result is established in Sobolev spaces, the homogenization problem in the space of functions of bounded variation being the object of \cite{BM}.