Calculus of Variations and Geometric Measure Theory

M. Gori - F. Maggi - P. Marcellini

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

created on 12 Apr 2002
modified on 06 Dec 2002


Published Paper

Inserted: 12 apr 2002
Last Updated: 6 dec 2002

Journal: Differential and Integral Equations
Volume: 16
Pages: 51-76
Year: 2003


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 2 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_{\textrm{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 \textit{Carathéodory} integrands too. More precisely, we assume that $f\left( x,s,\xi \right) $ is a nonnegative function, convex with respect to $\xi $, continuous in $\left(s,\xi \right) $ and such that the weak derivative $f_{x}$ is locally summable, i.e., $f(\cdot ,s,\xi )\in W_{\textrm{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 H\"{o}lder 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 1. Finally we investigate the lower semicontinuity of the integral $F(u,\Omega)$, with respect to the strong norm topology of $L^1_{\textrm{loc}}(\Omega)$, 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 $n\ge1$ and $m>1$.

1 Gori M., Marcellini P.: \textit{An extension of a Serrin's lower semicontinuity theorem}, J. Convex Analysis, \textbf{9 }% (2002), 1-28. 2 Serrin J.: \textit{On the definition and properties of certain variational integrals}, Trans. Amer. Math. Soc., \textbf{101 } (1961), 139-167.

Keywords: calculus of variations, Lower Semicontinuity, strong convergence in $L^1$