On minimizers of an isoperimetric problem with long-range interactions   under a convexity constraint
  We study a variational problem modeling the behavior at equilibrium of charged liquid drops under convexity constraint. After proving 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. 
Michael Goldman, Matteo Novaga, Berardo Ruffini
math.AP