Localization principle and relaxation

TL;DR

This paper establishes relaxation theorems for multiple integrals in Sobolev spaces under broad conditions, utilizing a localization principle to handle nonconvex integrals with exponential growth.

Contribution

It introduces a localization principle for relaxation theorems, extending results to nonconvex integrals with exponential growth in Sobolev spaces.

Findings

Proved relaxation theorems under general conditions.

Applied localization principle to nonconvex integrals.

Extended relaxation results to integrals with exponential growth.

Abstract

Relaxation theorems for multiple integrals on W^{1,p}(\Omega;\RR^m), where p\in]1,\infty[, are proved under general conditions on the integrand L:\MM\to[0,\infty] which is Borel measurable and not necessarily finite. We involve a localization principle that we previously used to prove a general lower semicontinuity result. We apply these general results to the relaxation of nonconvex integrals with exponential-growth.