Differential and Integral Equations

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

Gyula Csató

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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.

Article information

Source
Differential Integral Equations, Volume 28, Number 9/10 (2015), 971-988.

Dates
First available in Project Euclid: 23 June 2015

Permanent link to this document
https://projecteuclid.org/euclid.die/1435064546

Mathematical Reviews number (MathSciNet)
MR3360726

Zentralblatt MATH identifier
1363.49040

Subjects
Primary: 49Q20: Variational problems in a geometric measure-theoretic setting 30C35: General theory of conformal mappings 26D10: Inequalities involving derivatives and differential and integral operators

Citation

Csató, Gyula. An isoperimetric problem with density and the Hardy Sobolev inequality in $\mathbb R^2$. Differential Integral Equations 28 (2015), no. 9/10, 971--988. https://projecteuclid.org/euclid.die/1435064546


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