On some sharp conditions for lower semicontinuity in L

Let Ω be an open set of R and let f : Ω × R × R be a nonnegative continuous function, convex with respect to ξ ∈ R. Following the well known theory originated by Serrin [14] in 1961, we deal with the lower semicontinuity of the integral F (u,Ω) = ∫ Ω f (x, u(x), Du(x)) dx with respect to the Lloc (Ω) strong convergence. Only recently it has been discovered that dependence of f (x, s, ξ) on the x variable plays a crucial role in the lower semicontinuity. In this paper we propose a mild assumption on x that allows us to consider discontinuous integrands too. More precisely, we assume that f (x, s, ξ) is a nonnegative Carathéodory function, convex with respect to ξ, continuous in (s, ξ) and such that f(·, s, ξ) ∈ W 1,1 loc (Ω) for every s ∈ R and ξ ∈ R , with the L norm of fx(·, s, ξ) locally bounded. We also discuss some other conditions on x; in particular we prove that Hölder continuity of f with respect to x is not sufficient for lower semicontinuity, even in the one dimensional case, thus giving an answer to a problem posed by the authors in [12]. Finally we investigate the lower semicontinuity of the integral F (u,Ω), with respect to the strong norm topology of Lloc (Ω), in the vector-valued case, i.e., when f : Ω× R × Rm×n → R for some n ≥ 1 and m > 1.