Loss of double-integral character during relaxation

Abstract

We provide explicit examples to show that the relaxation of functionals $$L^p(\Omega ;{\mathbb{R}}^m)\ni u\mapsto {\int }_{\Omega }{\int }_{\Omega }W(u(x),u(y))dxdy,$$ where $\Omega \subset {\mathbb{R}}^n$ is an open and bounded set, $1<p<\infty$ and $W:{\mathbb{R}}^m\times {\mathbb{R}}^m\to \mathbb{R}$ 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.

arXiv comment22 pages, 1 figure