A simplified counterexample to the integral representation of the relaxation of double integrals

TL;DR

This paper presents a simplified counterexample demonstrating that the lower-semicontinuous envelope of a non-convex double integral cannot always be expressed as a double integral, challenging existing integral representation assumptions.

Contribution

It provides a straightforward counterexample and explicit formula for the relaxation, clarifying limitations in representing the lower-semicontinuous envelope of certain double integrals.

Findings

Counterexample shows non-representability as a double integral

Explicit relaxation formula derived for a specific integrand

Highlights limitations of Young measure characterization

Abstract

We show that the lower-semicontinuous envelope of a non-convex double integral may not admit a representation as a double integral. By taking an integrand with value $+\infty$ except at three points (say $-1$, $0$ and $1$) we give a simple proof and an explicit formula for the relaxation that hopefully may shed some light on this type of problems. This is a simplified version of examples by Mora-Corral and Tellini, and Kreisbeck and Zappale, who characterize the lower-semicontinuous envelope via Young measures.