An elementary proof of Acerbi Fusco minimizer existence theorem

TL;DR

This paper provides an elementary and accessible proof of the Acerbi-Fusco minimizer existence theorem, which is fundamental in calculus of variations, making the result more understandable for students with basic measure theory knowledge.

Contribution

The paper introduces a simplified proof of the theorem, avoiding advanced real and functional analysis tools, thus broadening accessibility for learners.

Findings

Proof is accessible to students with elementary measure theory.

The theorem's validity is established with a more straightforward approach.

Enhances understanding of weak lower semicontinuity in calculus of variations.

Abstract

The weak lower semicontinuity of the functional $$ F(u)=\int_{\Omega}f(x,u,\nabla u)\, dx$$ 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 proof that can be easily understood by students familiar only with the elementary measure theory.