Geodesic
An elementary proof of Marcellini Sbordone semicontinuity theorem
Kybernetika, Tome 59 (2023) no. 5, pp. 723-736.

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The weak lower semicontinuity of the functional $$ F(u)=\int_{\Omega}f(x,u,\nabla u) {\rm d} x $$ is a classical topic that was studied thoroughly. It was shown that if the function $f$ is continuous and convex in the last variable, the functional is sequentially weakly lower semicontinuous on $W^{1,p}(\Omega)$. However, the known proofs use advanced instruments of real and functional analysis. Our aim here is to present a proof understandable even for students familiar only with the elementary measure theory.
DOI : 10.14736/kyb-2023-5-0723
Classification : 46E35, 49J20, 49J45
Keywords: convexity; sequential semicontinuity; calculus of variation; minimizer 
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Roskovec, Tomáš G.; Soudský, Filip. An elementary proof of Marcellini Sbordone semicontinuity theorem. Kybernetika, Tome 59 (2023) no. 5, pp. 723-736. doi : 10.14736/kyb-2023-5-0723. http://geodesic.mathdoc.fr/articles/10.14736/kyb-2023-5-0723/

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