Calculus of Variations and Geometric Measure Theory

A. Pratelli - G. Saracco

On the isoperimetric problem with double density

created by saracco on 09 Apr 2018
modified on 15 Feb 2020


Published Paper

Inserted: 9 apr 2018
Last Updated: 15 feb 2020

Journal: Nonlinear Anal.
Volume: 177
Number: B
Pages: 733--752
Year: 2018
Doi: 10.1016/

ArXiv: 1804.02966 PDF

A subscript $r$ is missing in the hypothesis of Theorem A and related Lemmas in the published version. This version contains the correct statements.


In this paper we consider the isoperimetric problem with double density in an Euclidean space, that is, we study the minimisation of the perimeter among subsets of $\mathbb{R}^n$ with fixed volume, where volume and perimeter are relative to two different densities. The case of a single density, or equivalently, when the two densities coincide, has been well studied in the last years; nonetheless, the problem with two different densities is an important generalisation, also in view of applications. We will prove the existence of isoperimetric sets in this context, extending the known results for the case of single density.

Keywords: anisotropic perimeter, isoperimetric problem