## Differential and Integral Equations

### Renormalized variational principles and Hardy-type inequalities

#### Abstract

Let $\Omega\subset{{\mathbb{R}}}^2$ be a bounded domain on which Hardy's inequality holds. We prove that $[\exp(u^2)-1]/\delta^2\in L^1(\Omega)$ if $u\in H^1_0(\Omega)$, where $\delta$ denotes the distance to $\partial\Omega$. The corresponding higher-dimensional result is also given. These results contain both Hardy's and Trudinger's inequalities, and yield a new variational characterization of the maximal solution of the Liouville equation on smooth domains, in terms of a renormalized functional. A global $H^1$ bound on the difference between the maximal solution and the first term of its asymptotic expansion follows.

#### Article information

Source
Differential Integral Equations, Volume 19, Number 4 (2006), 437-448.

Dates
First available in Project Euclid: 21 December 2012