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 V is always isoperimetric for its own volumes (which may be smaller than V). In particular, if it is strictly smaller, we provide an explicit formula. Mathematics Subject Classification (2020). Primary: 49Q10. Secondary: 49Q20, 58B20