Calculus of Variations and Geometric Measure Theory

C. Kreisbeck - E. Zappale

Loss of double-integral character during relaxation

created on 31 Jul 2019
modified on 11 Oct 2020


Accepted Paper

Inserted: 31 jul 2019
Last Updated: 11 oct 2020

Journal: SIAM SIMA
Year: 2019

ArXiv: 1907.13180 PDF


We provide explicit examples to show that the relaxation of functionals $$ Lp(\Omega) \ni u\mapsto \int\Omega\int\Omega W(u(x), u(y))\, dx\, dy, $$ where $\Omega\subset\mathbb{R}^n$ is an open and bounded set, $1<p<\infty$ and $W:\mathbb{R}\times \mathbb{R}\to \mathbb{R}$ a suitable integrand, is in general not of double-integral form. This proves an up to now open statement in Pedregal, Rev. Mat. Complut. 29 (2016) and Bellido & Mora-Corral, SIAM J. Math. Anal. 50 (2018). The arguments are inspired by recent results regarding the structure of (approximate) nonlocal inclusions, in particular, their invariance under diagonalization of the constraining set. For a complementary viewpoint, we also discuss a class of double-integral functionals for which relaxation is in fact structure preserving and the relaxed integrands arise from separate convexification.

Keywords: relaxation, Lower Semicontinuity, nonlocality, double-integrals, nonlocal inclusions