Lower semicontinuity via W^{1,q}-quasiconvexity

TL;DR

This paper establishes a general localization principle under which W^{1,q}-quasiconvexity guarantees the sequential weak lower semicontinuity of certain integral functionals, with applications in calculus of variations.

Contribution

It introduces a new localization principle that links W^{1,q}-quasiconvexity to lower semicontinuity under broad conditions.

Findings

W^{1,q}-quasiconvexity ensures lower semicontinuity under the localization principle.

The paper provides applications demonstrating the principle's utility.

The results extend the understanding of lower semicontinuity in variational problems.

Abstract

We isolate a general condition, that we call "localization principle", on the integrand L:\MM\to[0,\infty], assumed to be continuous, under which W^{1,q}-quasiconvexity with q\in[1,\infty] is a sufficient condition for I(u)=\int_\Omega L(\nabla u(x))dx to be sequentially weakly lower semicontinuous on W^{1,p}(\Omega;\RR^m) with p\in]1,\infty[. Some applications are given.