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

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 C1,1-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.

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