A formula for the minimal perimeter of clusters with density

Abstract

This paper deals with the isoperimetric problem for clusters in a Euclidean space with double density. In particular, we show that a limit of an isoperimetric minimizing sequence of clusters with volumes $\mathbf{V}$ is always isoperimetric for its own volumes (which may be smaller than $\mathbf{V}$). In particular, if it is strictly smaller, we provide an explicit formula.