Large volume minimizers of a non local isoperimetric problem: theoretical and numerical approaches

We consider the volume-constrained minimization of the sum of the perimeter and the Riesz potential. We add an external potential of the form $\mathrm{\parallel }x{\mathrm{\parallel }}^{\beta }$ that provides the existence of a minimizer for any volume constraint, and we study the geometry of big minimizers. Then we provide a numerical method to adress this variational problem.