# Loss of double-integral character during relaxation

## TL;DR

This paper demonstrates that the relaxation of certain double-integral functionals in calculus of variations generally does not preserve the double-integral structure, resolving an open question in the field.

## Contribution

It provides explicit examples showing the loss of double-integral form upon relaxation, addressing an open problem in the mathematical analysis of nonlocal functionals.

## Key findings

- Relaxation of double-integral functionals often results in loss of double-integral structure.
- Explicit counterexamples are constructed to demonstrate this phenomenon.
- Some classes of functionals retain their structure after relaxation, linked to convexification.

## Abstract

We provide explicit examples to show that the relaxation of functionals $$ L^p(\Omega;\mathbb{R}^m) \ni u\mapsto \int_\Omega\int_\Omega W(u(x), u(y))\, dx\, dy, $$ 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.