Minimality of balls in the small volume regime for a general Gamow type functional

Abstract

We consider functionals given by the sum of the perimeter and the double integral of some kernel $g:{\mathbb{R}}^N\times {\mathbb{R}}^N\to {\mathbb{R}}^{+}$, multiplied by a "mass parameter" $\epsilon$. We show that, whenever $g$ is admissible, radial and decreasing, the unique minimizer of this functional among sets of given volume is the ball as soon as $\epsilon \ll 1$.