Calculus of Variations and Geometric Measure Theory

R. Cristoferi - G. Gravina

On the relaxation of functionals with contact terms on non-smooth domains

created by gravina on 20 Oct 2020
modified by cristoferi on 14 Sep 2021


Accepted Paper

Inserted: 20 oct 2020
Last Updated: 14 sep 2021

Journal: Indiana University Mathematics Journal
Year: 2021

ArXiv: 2010.03212 PDF


We provide the integral representation formula for the relaxation in $BV(\Omega; \mathbb{R}^M)$ with respect to strong convergence in $L^1(\Omega; \mathbb{R}^M)$ of a functional with a boundary contact energy term. This characterization is valid for a large class of surface energy densities, and for domains satisfying mild regularity assumptions. Motivated by some classical examples where lower semicontinuity fails, we analyze the extent to which the geometry of the set enters the relaxation procedure.