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.

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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