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

We prove, using elementary methods of complex analysis, the following generalization of the isoperimetric inequality: if $p\in \text{re}$, $\mathrm{\Omega }\subset {\text{re}}^2$ then the inequality $${\left(\frac{\mathrm{\mid }\mathrm{\Omega }\mathrm{\mid }}{\pi }\right)}^{\frac{p+1}{2}}\le \frac{1}{2\pi }{\int }_{\text{delomega}}\mathrm{\mid }x{\mathrm{\mid }}^{p}d\sigma (x)$$ holds true under appropriate assumptions on $\mathrm{\Omega }$ and $p\mathrm{.}$ This solves an open problem arising in the context of isoperimetric problems with density and poses some new ones (for instance generalizations to ${\text{re}}^n$). 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.