Note on the Lower Semicontinuity with Respect to the Weak Topology on $W^{1,p}(\Omega)$
Bollettino della Unione matematica italiana, Série 9, Tome 3 (2010) no. 2, pp. 381-390.

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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 $x \in \Omega$.
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Černý, Robert. Note on the Lower Semicontinuity with Respect to the Weak Topology on $W^{1,p}(\Omega)$. Bollettino della Unione matematica italiana, Série 9, Tome 3 (2010) no. 2, pp. 381-390. http://geodesic.mathdoc.fr/item/BUMI_2010_9_3_2_a8/

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