Calculus of Variations and Geometric Measure Theory

M. Goldman - M. Novaga - B. Ruffini

Rigidity of the ball for an isoperimetric problem with strong capacitary repulsion

created by goldman on 11 Jan 2022
modified by novaga on 29 Dec 2022


Accepted Paper

Inserted: 11 jan 2022
Last Updated: 29 dec 2022

Journal: J. Eur. Math. Soc.
Year: 2022

ArXiv: 2201.04376 PDF


We consider a variational problem involving competition between surface tension and charge repulsion. We show that, as opposed to the case of weak (short-range) interactions where we proved ill-posedness of the problem in a previous paper, when the repulsion is stronger the perimeter dominates the capacitary term at small scales. In particular we prove existence of minimizers for small charges as well as their regularity. Combining this with the stability of the ball under small $C^{1,\gamma}$ perturbations, this ultimately leads to the minimality of the ball for small charges. We cover in particular the borderline case of the $1-$capacity where both terms in the energy are of the same order.