Relaxation for partially coercive integral functionals with linear growth

TL;DR

This paper establishes an integral representation for the relaxation of certain integral functionals in BV spaces, accommodating unbounded integrands in the target variable, thus broadening the scope of variational analysis.

Contribution

It provides the first relaxation result for unbounded integrands in the $u$-variable, using a novel truncation approach and liftings theory, under general assumptions.

Findings

First relaxation theorem for unbounded integrands in $u$

Applicable to a broad class of partially coercive, quasiconvex integrands

Introduces a new truncation technique leveraging liftings theory

Abstract

We prove an integral representation theorem for the $\mathrm{L}^1(\Omega;\mathbb{R}^m)$-relaxation of the functional \[ \mathcal{F}\colon u\mapsto\int_\Omega f(x,u(x),\nabla u(x))\;\mathrm{dd } x,\quad u\in\mathrm{W}^{1,1}(\Omega;\mathbb{R}^m),\quad\Omega\subset\mathbb{R}^d\text{ open,} \] to the space $\mathrm{BV}(\Omega;\mathbb{R}^m)$ under very general assumptions, requiring principally that $f$ be Carath\'eodory, partially coercive, and quasiconvex in the final variable. Our result is the first of its kind which applies to integrands which are unbounded in the $u$-variable and thus allows to treat many problems from applications. Such functionals are out of reach of the classical blow-up approach introduced by Fonseca & M\"uller [Arch. Ration. Mech. Anal. 123 (1993), 1--49]. Our proof relies on an intricate truncation construction (in the $x$ and $u$ arguments simultaneously) made possible by the theory of liftings as introduced in the companion paper arXiv:1708.04165, and features techniques which could be of use for other problems featuring $u$-dependent integrands.