Lower semicontinuity and relaxation for free discontinuity functionals with non-standard growth

Abstract: A lower semicontinuity result and a relaxation formula for free discontinuity functionals with non-standard growth in the bulk energy are provided. Our analysis is based on a non-trivial adaptation of the blow-up (Ambrosio 1994) and of the global method for relaxation (Bouchitt\'e et al. 1998) to the setting of generalized special function of bounded variation with Orlicz growth. Key tools developed in this paper are an integral representation result and a Poincar\'e inequality under non-standard growth.