Calculus of Variations and Geometric Measure Theory

A. Braides

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

created by braidesa on 24 Apr 2023
modified on 30 Aug 2023


Accepted Paper

Inserted: 24 apr 2023
Last Updated: 30 aug 2023

Journal: Comptes Rendus - Série Mathématique
Year: 2023

ArXiv: 2212.11200 PDF
Links: ArXiv link


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.