Calculus of Variations and Geometric Measure Theory

M. Novaga - A. Pratelli

Minimisers of a general Riesz-type problem

created by novaga on 09 Jul 2020
modified on 29 Mar 2021


Published Paper

Inserted: 9 jul 2020
Last Updated: 29 mar 2021

Journal: Nonlinear Analysis
Volume: 209
Number: 112346
Year: 2021

ArXiv: 2007.02107 PDF


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 $g : \mathbb R^N\setminus\{0\}→\mathbb R^+$. 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 $g$.