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

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 \cite{BFLM} for a constant exponent. %Under the assumption of local log-H\"older continuity for the variable exponent $p(x)$, 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.