Manifold-constrained free discontinuity problems and Sobolev approximation

TL;DR

This paper investigates the regularity of local minimisers in a free-discontinuity problem constrained by a manifold and variable-exponent growth, extending Sobolev approximation results to establish regularity.

Contribution

It extends Sobolev approximation results to manifold-constrained free-discontinuity problems with variable exponents and proves regularity of local minimisers in this setting.

Findings

Extended Sobolev approximation to manifold-valued maps with small jump sets.

Proved regularity of local minimisers under the new setting.

Established foundational results for manifold-constrained free-discontinuity problems.

Abstract

We study the regularity of local minimisers of a prototypical free-discontinuity problem involving both a manifold-valued constraint on the maps (which are defined on a bounded domain $\Omega \subset \R^2$) and a variable-exponent growth in the energy functional. To this purpose, we first extend to this setting the Sobolev approximation result for special function of bounded variation with small jump set originally proved by Conti, Focardi, and Iurlano \cite{CFI-ARMA, CFI-AIHP} for special functions of bounded deformation. Secondly, we use this extension to prove regularity of local minimisers.