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

An integral representation result for free-discontinuity energies defined on the space $GSB{V}^{p(\cdot )}$ 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 $\mathrm{\Gamma }$-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.