Relaxation for Partially Coercive Integral Functionals with Linear Growth

Abstract

We prove an integral representation theorem for the L 1-relaxation of the functional F : u →ˆΩ f (x, u(x), ∇u(x)) dx, u ∈ W 1,1 (Ω; R m), where Ω ⊂ R d (d ≥ 2) is a bounded Lipschitz domain, to the space BV(Ω; R m) under very general assumptions: we require principally that f is Carathéodory, that the partial coercivity and linear growth bound g(x, y)|A| ≤ f (x, y, A) ≤ Cg(x, y)(1 + |A|), holds, where g : Ω × R m → [0, ∞) is a continuous function satisfying a weak monotonicity condition, and that f is quasiconvex in the final variable. Our result is the first that applies to integrands which are unbounded in the u-variable and therefore allows for the treatment of 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 developed in a previous paper by the authors [Arch. Ration. Mech. Anal., 232 (2019), 1227-1328], and features techniques which could be of use for other problems involving u-dependent integrands.