On the relaxation of unbounded multiple integrals

TL;DR

This paper investigates the relaxation of nonconvex multiple integrals in calculus of variations, providing integral representations in Sobolev spaces and applications to scalar and quasiconvex growth cases.

Contribution

It introduces local measure-based techniques to characterize the relaxation of unbounded, nonconvex integrals with convex domains, extending existing theories.

Findings

Established integral representation results in Sobolev spaces.

Extended relaxation theory to integrands with quasiconvex and p(x)-growth.

Provided applications in scalar and variable exponent cases.

Abstract

We study the relaxation of multiple integrals of the calculus of variations, where the integrands are nonconvex with convex effective domain and can take the value \infty. We use local techniques based on measure arguments to prove integral representation in Sobolev spaces of functions which are almost everywhere differentiable. Applications are given in the scalar case and in the case of integrands with quasiconvex growth and p(x)-growth.