On some sharp conditions for lower semicontinuity in \(L^1\). (English) Zbl 1028.49012

Summary: 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 J. Serrin [Trans. Am. Math. Soc. 101, 139-167 (1961; Zbl 0102.04601)], we deal with the lower semicontinuity of the integral \[ F(u,\Omega)= \int_\Omega \bigl(x,u(x), Du(x)\bigr)dx \] with respect to the \(L^1_{\text{loc}} (\Omega)\) strong convergence. Only recently it has been discovered that dependence of \(f(x,s,\xi)\) 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 \(f(x, s,\xi)\) is a nonnegative Carathéodory function, convex with respect to \(\xi\), continuous in \((s,\xi)\) and such that \(f(\cdot,s,\xi) \in W^{1,1}_{\text{loc}} (\Omega)\) 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 [J. Convex Anal. 9, No. 2, 475-502 (2002; Zbl 1019.49021)]. Finally, we investigate the lower semicontinuity of the integral \(F(u,\Omega)\), with respect to the strong norm topology of \(L^1_{\text{loc}} (\Omega)\), in the vector-valued case, i.e., when \(f:\Omega\times \mathbb{R}^m \times\mathbb{R}^{m\times n}\to \mathbb{R}\) for some \(n\geq 1\) and \(m>1\).


49J45 Methods involving semicontinuity and convergence; relaxation
49J10 Existence theories for free problems in two or more independent variables