Kybernetika 59 no. 5, 723-736, 2023

An elementary proof of Marcellini Sbordone semicontinuity theorem

Tomáš G. Roskovec and Filip SoudskýDOI: 10.14736/kyb-2023-5-0723

Abstract:

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.

Keywords:

convexity, sequential semicontinuity, calculus of variation, minimizer

Classification:

49J45, 49J20, 46E35

paper.pdf

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