RELAXATION FOR PARTIALLY COERCIVE INTEGRAL FUNCTIONALS WITH LINEAR GROWTH

We prove an integral representation theorem for the L-1-relaxation of the functional F: u bar right arrow integral(Omega) f(x,u(x),del u(x)) dx, u is an element of W-1,W-1 (Omega;R-m), where Omega subset of R-d (d >= 2) is a bounded Lipschitz domain, to the space BV(Omega;R-m) under very general assumptions: we require principally that f is Caratheodory, that the partial coercivity and linear growth bound g(x , y)vertical bar A vertical bar <= f(x, y, A) <= Cg(x,y)(1+ vertical bar A vertical bar), hold, where g: (Omega) over bar x R-m -> [0, infinity) is a continuous function satisfying a weak monotonicity condition, and that f is quasi-convex 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 blowup approach introduced by Fonseca and Muller [Arch. Ration. Mech. Anal., 123 (1993), pp. 1-49]. Our proof relies on an intricate truncation construction (in the x- and v.-arguments simultaneously) made possible by the theory of liftings developed in a previous paper by the authors [Arch. Ration. Mech. Anal., 232 (2019), pp. 1227-1328], and features techniques which could be of use for other problems involving u-dependent integrands.