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

TL;DR

This paper establishes lower semicontinuity and relaxation formulas for free discontinuity functionals with non-standard Orlicz growth, extending classical methods to more general growth conditions.

Contribution

It adapts blow-up and relaxation techniques to generalized special functions of bounded variation with Orlicz growth, providing new integral representation and Poincaré inequality results.

Findings

Derived a lower semicontinuity result for non-standard growth functionals

Established a relaxation formula in the Orlicz growth setting

Developed new integral representation and Poincaré inequality tools

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.