## Differential and Integral Equations

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

### 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.

#### 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