Minimisers of a general Riesz-type Problem

Abstract

We consider sets in 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 : R N \ {0} → 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.