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