A survey on the existence of isoperimetric sets in the space ${\mathbb{R}}^N$ with density

Abstract

The aim of this survey is to give a precise idea of the recent results on existence of isoperimetric sets in ${\mathbb{R}}^N$ with density. We will mainly focus on the overall ideas, leaving away some technical details of the proofs, which can be found in the cited papers. No previous knowledge on the subject is assumed from the reader. This survey originates from a talk of the author at the conference ‘‘New Trends in Nonlinear PDE’s’’ held at the Accademia dei Lincei on November 26th, 2013. I wish to dedicate this paper to Carlo Sbordone, because of his recent 65th birthday, and to Ula, because she will become my wife in a few days.