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September/October 2015 An isoperimetric problem with density and the Hardy Sobolev inequality in $\mathbb R^2$
Gyula Csató
Differential Integral Equations 28(9/10): 971-988 (September/October 2015).

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.

Citation

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Gyula Csató. "An isoperimetric problem with density and the Hardy Sobolev inequality in $\mathbb R^2$." Differential Integral Equations 28 (9/10) 971 - 988, September/October 2015.

Information

Published: September/October 2015
First available in Project Euclid: 23 June 2015

zbMATH: 1363.49040
MathSciNet: MR3360726

Subjects:
Primary: 26D10, 30C35, 49Q20

Rights: Copyright © 2015 Khayyam Publishing, Inc.

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Vol.28 • No. 9/10 • September/October 2015
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