An Isoperimetric Problem With Density and the Hardy Sobolev Inequality in $\mathbb{R}^2$

TL;DR

This paper generalizes the isoperimetric inequality with density in  using complex analysis, establishes its equivalence with a Hardy-Sobolev inequality, and explores implications for the harmonic transplantation method.

Contribution

It introduces a new isoperimetric inequality with density, proves its equivalence to a Hardy-Sobolev inequality, and extends classical methods to singular embeddings.

Findings

Proved a generalized isoperimetric inequality with density in .

Established the best constant in the Hardy-Sobolev inequality.

Linked the inequality to the harmonic transplantation method.

Abstract

We prove, using elementary methods of complex analysis, the following generalization of the isoperimetric inequality: if $p\in\re$, $\Omega\subset\re^2$ then the inequality $$ \left(\frac{|\Omega|}{\pi}\right)^{\frac{p+1}{2}}\leq\frac{1}{2\pi}\int_{\delomega}|x|^pd\sigma(x) $$ holds true under appropriate assumptions on $\Omega$ and $p.$ This solves an open problem arising in the context of isoperimetric problems with density and poses some new ones (for instance generalizations to $\re^n$). We prove the equivalence with a Hardy-Sobolev inequality, giving the best constant, and generalize thereby the equivalence between the classical isoperimetric inequality and the Sobolev inequality. Furthermore, the inequality paves the way for solving another problem: the generalization of the harmonic transplantation method of Flucher to the singular Moser-Trudinger embedding.