Regularity of minimizers for free-discontinuity problems with $p(\cdot )$-growth

A regularity result for free-discontinuity energies defined on the space $SB{V}^{p(\cdot )}$ of special functions of bounded variation with variable exponent is proved, under the assumption of a log-Hölder continuity for the variable exponent $p(x)$. Our analysis expand on the regularity theory for minimizers of a class of free-discontinuity problems in the nonstandard growth case. This may be seen as a follow-up of the paper Fusco, Mingione and Trombetti (2001), dealing with a constant exponent.