On existence and nonexistence of isoperimetric inequality with differents monomial weights

Abstract

We consider the monomial weight $x^A=|x_1{|}^{a_1}\dots |x_N{|}^{a_N}$, where $a_i$ is a nonnegative real number for each $i\in \{1,\dots ,N\}$, and we establish the existence and nonexistence of isoperimetric inequalities with different monomial weights. We study positive minimizers of ${\int }_{\partial \Omega }{x}^A{\mathcal{H}}^{N-1}(x)$ among all smooth bounded sets $\Omega$ in ${\mathbb{R}}^N$ with fixed Lebesgue measure with monomial weight ${\int }_{\Omega }{x}^{B}dx$.

References in corpus (2)

• Isoperimetric Regions in ${\mathbb{R}}^n$ with density $r^p$
• Balls Isoperimetric in ${\mathbb{R}}^n$ with Volume and Perimeter Densities $r^m$ and $r^k$

Cited by in corpus (1)

• The Cheeger problem in abstract measure spaces