On the Existence and Nonexistence of Isoperimetric Inequalities with Different Monomial Weights

Abstract

We consider the monomial weight \(x^{A}=\vert x_{1}\vert ^{a_{1}}\cdots \vert x_{N}\vert ^{a_{N}}\), 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 \(\int _{\partial \Omega }x^{A}d\mathcal {H}^{N-1}(x)\) among all smooth bounded open sets \(\Omega \) in \({\mathbb {R}}^{N}\) with fixed Lebesgue measure and monomial weight \(\int _{\Omega }x^{B}dx\). Besides that, we also establish a weighted perimeter inequality under a new version of Steiner Symmetrization.