# An elementary proof of Marcellini Sbordone semicontinuity theorem

Kybernetika (2023)

• Volume: 59, Issue: 5, page 723-736
• ISSN: 0023-5954

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## Abstract

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The weak lower semicontinuity of the functional $F\left(u\right)={\int }_{\Omega }f\left(x,u,\nabla u\right)\mathrm{d}x$ 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 ${W}^{1,p}\left(\Omega \right)$. 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.

## How to cite

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Roskovec, Tomáš G., and Soudský, Filip. "An elementary proof of Marcellini Sbordone semicontinuity theorem." Kybernetika 59.5 (2023): 723-736. <http://eudml.org/doc/299166>.

@article{Roskovec2023,
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.},
author = {Roskovec, Tomáš G., Soudský, Filip},
journal = {Kybernetika},
keywords = {convexity; sequential semicontinuity; calculus of variation; minimizer},
language = {eng},
number = {5},
pages = {723-736},
publisher = {Institute of Information Theory and Automation AS CR},
title = {An elementary proof of Marcellini Sbordone semicontinuity theorem},
url = {http://eudml.org/doc/299166},
volume = {59},
year = {2023},
}

TY - JOUR
AU - Roskovec, Tomáš G.
AU - Soudský, Filip
TI - An elementary proof of Marcellini Sbordone semicontinuity theorem
JO - Kybernetika
PY - 2023
PB - Institute of Information Theory and Automation AS CR
VL - 59
IS - 5
SP - 723
EP - 736
AB - 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.
LA - eng
KW - convexity; sequential semicontinuity; calculus of variation; minimizer
UR - http://eudml.org/doc/299166
ER -

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