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

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.