An isoperimetric problem with density and the Hardy Sobolev inequality in $\mathbb R^2$

Abstract

We prove, using elementary methods of complex analysis, the following generalization of the isoperimetric inequality: if $p\in\mathbb{R}$, $\Omega\subset\mathbb{R}^2$, then the inequality $$ \big (\frac{|\Omega|}{\pi} \big )^{\frac{p+1}{2}} \leq\frac{1}{2\pi}\int_{\partial\Omega}|x|^pd\sigma(x), $$ 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 $\mathbb{R}^n$). 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.