Calculus of Variations and Geometric Measure Theory

S. Almi - D. Reggiani - F. Solombrino

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

created by almi1 on 17 Jan 2023
modified by solombrino on 22 Dec 2023


Published Paper

Inserted: 17 jan 2023
Last Updated: 22 dec 2023

Journal: Calculus of Variations and Partial Differential Equations
Volume: 63
Number: 1
Pages: art. 24
Year: 2024
Doi: 10.1007/s00526-023-02623-2

ArXiv: 2301.07406 PDF


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 of Ambrosio (1994) and of the global method for relaxation of Bouchitté-Fonseca-Leoni-Mascarenhas (2002) 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 Poincaré inequality under non-standard growth.