Loss of double-integral character during relaxation

We provide explicit examples to show that the relaxation of functionals $L^p(Ω;\mathbb{R}^m) \ni u\mapsto \int_Ω\int_ΩW(u(x), u(y))\, dx\, dy,$ where $Ω\subset\mathbb{R}^n$ 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.

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