*Accepted Paper*

**Inserted:** 31 jul 2019

**Last Updated:** 11 oct 2020

**Journal:** SIAM SIMA

**Year:** 2019

**Abstract:**

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

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