#### Vol. 11, No. 5, 2018

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On minimizers of an isoperimetric problem with long-range interactions under a convexity constraint

### Michael Goldman, Matteo Novaga and Berardo Ruffini

Vol. 11 (2018), No. 5, 1113–1142
##### Abstract

We study a variational problem modeling the behavior at equilibrium of charged liquid drops under a convexity constraint. After proving the well-posedness of the model, we show ${C}^{1,1}$-regularity of minimizers for the Coulombic interaction in dimension two. As a by-product we obtain that balls are the unique minimizers for small charge. Eventually, we study the asymptotic behavior of minimizers, as the charge goes to infinity.

##### Keywords
nonlocal isoperimetric problem, convexity constraint
##### Mathematical Subject Classification 2010
Primary: 49J30, 49J45, 49S05
##### Milestones
Received: 8 November 2016
Revised: 6 July 2017
Accepted: 2 January 2018
Published: 11 April 2018
##### Authors
 Michael Goldman Université Paris-Diderot Sorbonne Paris-Cité Sorbonne Université, CNRS Laboratoire Jacques-Louis Lions Paris France Matteo Novaga Dipartimento di Matematica Università di Pisa Pisa Italy Berardo Ruffini Institut Montpelliérain Alexander Grothendieck University of Montpellier, CNRS Montpellier France