Integral representation and G-convergence for free-discontinuity problems with p(.)-growth

An integral representation result for free-discontinuity energies defined on the space GSBV(p(.)) of generalized special functions of bounded variation with variable exponent is proved, under the assumption of log-H & ouml;lder continuity for the variable exponent p(x). Our analysis is based on a variable exponent version of the global method for relaxation devised in Bouchitt & eacute; et al. (Arch Ration Mech Anal 165:187-242, 2002) for a constant exponent. We prove G-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.

Automated summary

An integral representation result for free-discontinuity energies on the space GSBV(p(.)) with variable exponent p(x) under the assumption of log-Hölder continuity is proved. The analysis is based on a variable exponent version of the global method for relaxation devised by Bouchitté et al. (Arch Ration Mech Anal 165:187-242, 2002) for a constant exponent. G-convergence of sequences of energies of the same type is proven, limit integrands are identified using asymptotic cell formulas, and a non-interaction property between bulk and surface contributions is proved.