Strong existence for free-discontinuity problems with non-standard growth

TL;DR

This paper establishes regularity results for minimizers of free-discontinuity problems with non-standard growth conditions, expanding the theoretical understanding of such variational problems in generalized Orlicz spaces.

Contribution

It provides the first Ahlfors-type regularity result for free-discontinuity energies in $SBV^{}$ with $$-growth, broadening the scope of regularity theory.

Findings

Proves Ahlfors-type regularity for free-discontinuity energies.

Extends regularity theory to non-standard growth conditions.

Analyzes minimizers in generalized Orlicz spaces.

Abstract

An Ahlfors-type regularity result for free-discontinuity energies defined on the space $SBV^{\varphi}$ of special functions of bounded variation with $\varphi$-growth, where $\varphi$ is a generalized Orlicz function, is proved. Our analysis expands on the regularity theory for minimizers of a class of free-discontinuity problems in the non-standard growth case.