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

created by scilla on 20 Apr 2022
modified on 25 Apr 2022

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Submitted Paper

Inserted: 20 apr 2022
Last Updated: 25 apr 2022

Year: 2022

ArXiv: 2204.09530 PDF

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-Hölder continuity for the variable exponent $p(x)$. Our analysis is based on a variable exponent version of the global method for relaxation devised in Bouchitté, 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.