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arXiv · 2204.09530

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

Giovanni Scilla·Francesco Solombrino·Bianca Stroffolini
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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.

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Giovanni Scilla, Francesco Solombrino, Bianca Stroffolini. 2022-04-20. Integral representation and $\Gamma$-convergence for free-discontinuity problems with $p(\cdot)$-growth. https://arxiv.org/abs/2204.09530

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