Calculus of Variations and Geometric Measure Theory

G. Scilla - F. Solombrino - B. Stroffolini

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

created by scilla on 20 Apr 2022
modified on 18 Aug 2023


Published Paper

Inserted: 20 apr 2022
Last Updated: 18 aug 2023

Journal: Calculus of Variations and Partial Differential Equations
Volume: 62
Number: 213
Year: 2023
Doi: 10.1007/s00526-023-02549-9

ArXiv: 2204.09530 PDF


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.

Keywords: Free-discontinuity problems, Integral representation, Γ-convergence, p(x)-growth