On some sharp conditions for lower semicontinuity in $L^1$

Abstract

Let $\Omega $ be an open set of $\mathbb{R}^{n}$ and let $f:\Omega \times \mathbb{R}\times \mathbb{R}^{n}$ be a nonnegative continuous function, convex with respect to $\xi \in \mathbb{R}^{n}$. Following the well known theory originated by Serrin Serrin [14] in 1961, we deal with the lower semicontinuity of the integral $F\left( u,\Omega \right) =\int_{\Omega }f\left( x,u(x),Du(x)\right) \,dx$ with respect to the $L_{\text{loc} }^{1}\left( \Omega \right) $ strong convergence. Only recently it has been discovered that dependence of $f\left( x,s,\xi \right) $ 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\left( x,s,\xi \right) $ is a nonnegative Carathéodory function, convex with respect to $\xi $ , continuous in $\left( s,\xi \right) $ and such that $f(\cdot ,s,\xi )\in W_{\text{loc}}^{1,1}\left( \Omega \right) $ for every $s\in \mathbb{R}$ and $\xi \in \mathbb{R}^{n}$, with the $L^{1}$ norm of $f_{x}(\cdot ,s,\xi )$ 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\left( u,\Omega \right) $, with respect to the strong norm topology of $L_{\text{ loc}}^{1}\left( \Omega \right) $, in the vector-valued case, i.e., when $ f:\Omega \times \mathbb{R}^{m}\times \mathbb{R}^{m\times n}\rightarrow \mathbb{R}$ for some $n\geq 1$ and $m>1$.