On the isoperimetric problem with perimeter density r^p

Abstract

In this paper the author studies the isoperimetric problem in $\re^n$ with perimeter density $|x{|}^p$ and volume density $1.$ We settle completely the case $n=2,$ completing a previous work by the author: we characterize the case of equality if $0\le p\le 1$ and deal with the case $-\infty <p<-1$ (with the additional assumption $0\in \Omega$). In the case $n\ge 3$ we deal mainly with the case $-\infty <p<0,$ showing among others that the results in $2$ dimensions do not generalize for the range $-n+1<p<0.$

References in corpus (2)

• The singular Moser- Trudger Inequality on simply connected domains
• Balls Isoperimetric in ${\mathbb{R}}^n$ with Volume and Perimeter Densities $r^m$ and $r^k$