On existence and nonexistence of isoperimetric inequality with differents monomial weights

TL;DR

This paper investigates the conditions under which isoperimetric inequalities with monomial weights exist or do not exist, focusing on positive minimizers for weighted perimeter and volume in Euclidean space.

Contribution

It establishes criteria for the existence and nonexistence of isoperimetric inequalities involving different monomial weights, advancing understanding of weighted geometric inequalities.

Findings

Identifies conditions for existence of weighted isoperimetric inequalities.

Provides nonexistence results under certain weight configurations.

Characterizes minimizers for weighted perimeter and volume problems.

Abstract

We consider the monomial weight $x^{A}=\vert x_{1}\vert^{a_{1}}\ldots\vert x_{N}\vert^{a_{N}}$, where $a_{i}$ is a nonnegative real number for each $i\in\{1,\ldots,N\}$, 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 $\int_{\Omega}x^{B}dx$.