Integral representation and $\Gamma$-convergence for free-discontinuity problems with $p(\cdot)$-growth

TL;DR

This paper establishes an integral representation and $ ext{Γ}$-convergence results for free-discontinuity problems involving variable exponent growth, extending classical methods to a more general setting with variable exponents.

Contribution

It introduces a variable exponent version of the relaxation method for free-discontinuity energies, providing new integral representation and $ ext{Γ}$-convergence results in this context.

Findings

Proved integral representation for energies in $GSBV^{p(ullet)}$.

Established $ ext{Γ}$-convergence of energy sequences.

Identified limit integrands via asymptotic cell formulas.

Abstract

An integral representation result for free-discontinuity energies defined on the space $GSBV^{p(\cdot)}$ of generalized special functions of bounded variation with variable exponent is proved, under the assumption of log-H\"older continuity for the variable exponent $p(x)$. Our analysis is based on a variable exponent version of the global method for relaxation devised in Bouchitt\`e, Fonseca, Leoni and Mascarenhas (2002) for a constant exponent. We prove $\Gamma$-convergence of sequences of energies of the same type, we identify the limit integrands in terms of asymptotic cell formulas and prove a non-interaction property between bulk and surface contributions.