Renormalized variational principles and Hardy-type inequalities

TL;DR

This paper establishes new inequalities combining Hardy and Trudinger types, providing a variational characterization of solutions to the Liouville equation and bounds on their asymptotic behavior.

Contribution

It introduces a renormalized variational principle that unifies Hardy and Trudinger inequalities and applies to characterize maximal solutions of the Liouville equation.

Findings

Proves integrability of a renormalized exponential function involving Hardy's inequality.

Provides higher-dimensional generalizations of the inequalities.

Derives bounds on the difference between solutions and their asymptotic expansion.

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.