ALEX 2018 Workshop: Abstracts

Relaxation of nonlocal supremal functionals

Carolin Kreisbeck${}^{(1)}$ and Elvira Zappale${}^{\mathrm{(}\mathrm{2}\mathrm{)}}$

(1) Universiteit Utrecht, Mathematisch Instituut (The Netherlands)

(2) Università degli Studi di Salerno, Dipartimento di Ingegneria Industriale (Italy)

Nonlocal functionals in the form of double integrals appear naturally in models of peridynamics. In the homogeneous case, separate convexity of the integrands has been identified as a necessary and sufficient condition for lower semicontinuity [5, 2, 4]. When it comes to relaxation, though, a characterization of the lower semicontinuous envelopes is still largely open. Indeed, in contrast to what one would expect, simple examples in [2, 1] indicate that the relaxed functionals do not follow from separate convexification, and hence, it is unclear whether they can be represented as double integrals.

Motivated by these recent developments, this talk addresses a related question by discussing homogeneous supremal functionals in the nonlocal setting, precisely,

$L^{\mathrm{\infty }}(\mathrm{\Omega };{ℝ}^m)\ni u\mapsto {\mathrm{esssup}}_{(x,y)\in \mathrm{\Omega }\times \mathrm{\Omega }}W(u(x),u(y)),$

with $\mathrm{\Omega }\subset {ℝ}^n$ a bounded, open set and a continuous density $W:{ℝ}^m\times {ℝ}^m\to ℝ$. We show that weak${}^{\ast }$ lower semicontinuity holds if and only if the level sets of a symmetrized and suitably diagonalized version of $W$ are separately convex. It turns out that, unlike for double integrals, the supremal structure of the functionals we consider here is guaranteed to be preserved in the process of relaxation. The proof of this statement relies on the connection between supremal and indicator functionals, which reduces the problem to studying weak${}^{\ast }$ closures of a class of non-local inclusions. We give examples of explicit relaxation formulas for different multi-well functions.