Minimisers of a general Riesz-type problem (Q2033039)

The authors consider sets in \(\mathbb {R}^N\) which minimize, for fixed volume, the sum of the perimeter and a non-local term given by the double integral of a kernel \[g : \mathbb {R}^N\setminus {0} \to \mathbb {R}^+.\] They establish some general existence and regularity results for minimizers. In the two-dimensional case they show that balls are the unique minimizers in the perimeter-dominated regime, for a wide class of functions \(g\).