Relaxation for partially coercive integral functionals with linear growth

Abstract

We prove an integral representation theorem for the ${\mathrm{L}}^1(\Omega ;{\mathbb{R}}^m)$-relaxation of the functional \[ \mathcal{F}\colon u\mapsto\int_Ωf(x,u(x),\nabla u(x))\;\mathrm{dd } x,\quad u\in\mathrm{W}^{1,1}(Ω;\mathbb{R}^m),\quadΩ\subset\mathbb{R}^d\text{ open,} \] to the space $\mathrm{BV}(\Omega ;{\mathbb{R}}^m)$ under very general assumptions, requiring principally that $f$ be Carathéodory, 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 & Müller [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.

48 pages. Updated to correct minor typos and inaccuracies, expanded exposition for clarity. Revised statement of Theorem 2.3, and a full proof is now given. Significant refactoring/rewriting of the proof of Proposition 4.1 (which was previously Proposition 4.2) for clarity. A proof is now given for Lemma 5.5. arXiv admin note: text overlap with arXiv:1708.04165