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

Černý, Robert

Note on the Lower Semicontinuity with Respect to the Weak Topology on

W^{1,p}(\Omega)

Černý, Robert

Bollettino dell'Unione Matematica Italiana, Tome 3 (2010), p. 381-390 / Harvested from Biblioteca Digitale Italiana di Matematica

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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 $W^{1,p}(\Omega)$ , $1\leq p\leq\infty$ , if and only if $g$ is convex in the second variable for almost every $x\in\Omega$ .

Publié le : 2010-06-01

MR 2666365 | Zbl 1196.49010

@article{BUMI_2010_9_3_2_381_0,
     author = {Robert \v Cern\'y},
     title = {Note on the Lower Semicontinuity with Respect to the Weak Topology on $W^{1,p}(\Omega)$},
     journal = {Bollettino dell'Unione Matematica Italiana},
     volume = {3},
     year = {2010},
     pages = {381-390},
     zbl = {1196.49010},
     mrnumber = {2666365},
     language = {en},
     url = {http://dml.mathdoc.fr/item/BUMI_2010_9_3_2_381_0}
}

Černý, Robert. Note on the Lower Semicontinuity with Respect to the Weak Topology on $W^{1,p}(\Omega)$. Bollettino dell'Unione Matematica Italiana, Tome 3 (2010) pp. 381-390. http://gdmltest.u-ga.fr/item/BUMI_2010_9_3_2_381_0/

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