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

Abstract An integral representation result for free-discontinuity energies defined on the space $$GSBV^{p(\cdot )}$$ G S B V p ( · ) 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 et al. (Arch Ration Mech Anal 165:187–242, 2002) for a constant exponent. We prove $$\Gamma $$ Γ -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.