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An integral representation result for free-discontinuity energies defined onthe space $GSBV^{p(\cdot)}$ of generalized special functions of boundedvariation with variable exponent is proved, under the assumption oflog-H\"older continuity for the variable exponent $p(x)$. Our analysis is basedon a variable exponent version of the global method for relaxation devised inBouchitt\`e, Fonseca, Leoni and Mascarenhas (2002) for a constant exponent. Weprove $\Gamma$-convergence of sequences of energies of the same type, weidentify the limit integrands in terms of asymptotic cell formulas and prove anon-interaction property between bulk and surface contributions.

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