Inserted: 31 jul 2019
Last Updated: 31 jul 2019
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