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

created by csato on 17 Mar 2015

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Submitted Paper

Inserted: 17 mar 2015
Last Updated: 17 mar 2015

Year: 2015
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 $\left(\frac{\vert\Omega\vert}{\pi}\right)^{\frac{p+1}{2}}\leq\frac{1}{2\pi}\int_{\partial\Omega}\vert x\vert ^pd\sigma(x)$ 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.