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

TL;DR

This paper derives an integral representation for the relaxed form of functionals with boundary contact terms in BV spaces, considering domain geometry and surface energy densities, addressing issues of lower semicontinuity.

Contribution

It provides a general relaxation formula for functionals with boundary contact energies in BV spaces, accounting for domain geometry and surface energy densities.

Findings

Integral representation formula for relaxed functionals with boundary contact terms.

Analysis of how domain geometry affects relaxation and lower semicontinuity.

Applicable to a broad class of surface energy densities and domains.

Abstract

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.