Geodesic
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 Cet article a éte moissonné depuis la source Biblioteca Digitale Italiana di Matematica

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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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     author = {\v{C}ern\'y, Robert},
     title = {Note on the {Lower} {Semicontinuity} with {Respect} to the {Weak} {Topology} on $W^{1,p}(\Omega)$},
     journal = {Bollettino della Unione matematica italiana},
     pages = {381--390},
     year = {2010},
     volume = {Ser. 9, 3},
     number = {2},
     zbl = {1196.49010},
     mrnumber = {2666365},
     language = {en},
     url = {http://geodesic.mathdoc.fr/item/BUMI_2010_9_3_2_a8/}
}
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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/