Minimisers of a general Riesz-type Problem

Abstract

We consider sets in ${\mathbb{R}}^N$ which minimise, for fixed volume, the sum of the perimeter and a non-local term given by the double integral of a kernel $g:{\mathbb{R}}^N\setminus \{0\}\to {\mathbb{R}}^{+}$. We establish some general existence and regularity results for minimisers. In the two-dimensional case we show that balls are the unique minimisers if the perimeter-dominated regime, for a wide class of functions $g$.

29 pages, 4 figures