ETD

• almost minimizers of the perimeter
• Gamow model
• non-local isoperimetric problem
• Riesz-type energy

We study the functional \[\mathcal F(E)=\mathcal P(E)+\epsilon\int_E\int_Eg(x-y)dxdy,\] defined on Borel sets of $\R^N$, where $g:\R^N\to[0,+\infty]$ is radially decreasing, continuous and finite outside of the origin and in $L^1_{loc}$, $\mathcal P$ is the perimeter functional and $\epsilon>0$ is a parameter. For every fixed $\epsilon$ we prove the existence of minimizers under a small volume constraint, their connectedness and the partial regularity of their boundary. Then we fix a mass constraint and we prove that for $\epsilon$ small enough the only possible minimizers are balls if $g$ is also radial. These results were already known for $g(x)=|x|^{-\lambda}$ with $0<\lambda<N$ being an assigned constant, and we are able to extend them. With our results we also treat the case with $\mathcal P_s$ instead of $\mathcal P$, and using some different ideas we can replace the perimeter with the power-like attractive term $\int_E\int_E|x-y|^{\alpha}\dx\dy$ for $\alpha>0$.