On an isoperimetric problem with a competing non-local term: quantitative results

This paper provides a quantitative version of the recent result of Knüpfer and Muratov (Commun Pure Appl Math 66:1129–1162, 2013) concerning the solutions of an extension of the classical isoperimetric problem in which a non-local repulsive term involving Riesz potential is present. In that work, it was shown that in two space dimensions the minimizer of the considered problem is either a ball or does not exist, depending on whether or not the volume constraint lies in an explicit interval around zero, provided that the Riesz kernel decays sufficiently slowly. Here, we give an explicit estimate for the exponents of the Riesz kernel for which the result holds.