Note on the Lower Semicontinuity with Respect to the Weak Topology on $W^{1,p}(\Omega)$

Černý, Robert

Bollettino della Unione matematica italiana, Série 9, Tome 3 (2010) no. 2, pp. 381-390

Résumé

Let $\Omega \subset \mathbf{R}^{N}$ be an open bounded set with a Lipschitz boundary and let $g: \Omega \times \mathbf{R} \to \mathbf{R}$ be a Carathéodory function satisfying usual growth assumptions. Then the functional $$\Phi(u) = \int_{\Omega} g(x,u(x)) \, dx$$ is lower semicontinuous with respect to the weak topology on $W^{1,p}(\Omega)$, $1 \le p \le \infty$, if and only if $g$ is convex in the second variable for almost every $x \in \Omega$.