Homogenization of variational problems in manifold valued BV-spaces

TL;DR

This paper extends homogenization results for variational problems to manifold-valued BV spaces, identifying the effective energy through Gamma-convergence, with applications to bulk, Cantor, and jump parts involving manifold geometry.

Contribution

It introduces a homogenization framework for integral functionals on BV maps into manifolds, including a novel characterization of the jump part via geodesic problems.

Findings

Homogenized energy is finite for BV maps into the manifold.

Bulk and Cantor parts involve tangential homogenized density.

Jump part characterized by a geodesic-based surface density.

Abstract

This paper extends the result of \cite{BM} on the homogenization of integral functionals with linear growth defined for Sobolev maps taking values in a given manifold. Through a $\Gamma$-convergence analysis, we identify the homogenized energy in the space of functions of bounded variation. It turns out to be finite for $BV$-maps with values in the manifold. The bulk and Cantor parts of the energy involve the tangential homogenized density introduced in \cite{BM}, while the jump part involves an homogenized surface density given by a geodesic type problem on the manifold.