Regularity of minimizers for free-discontinuity problems with p(<middle dot>)-growth

A regularity result for free-discontinuity energies defined on the space SBVp(<middle dot>) of special functions of bounded variation with variable exponent is proved, under the assumption of a log-Holder continuity for the variable exponent p(x). Our analysis expands 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 N. Fusco et al., J. Convex Anal. 8 (2001) 349-367, dealing with a constant exponent.