Renormalized variational principles and Hardy-type inequalities

Abstract

Let $\hat{I}\text{c◯}\subset {\mathbb{R}}^2$ be a bounded domain on which Hardy's inequality holds. We prove that $[\exp (u^2)-1]/\hat{I}{´}^2\in L^1(\hat{I}\text{c◯})$ if $u\in H_0^1(\hat{I}\text{c◯})$, where $\hat{I}´$ denotes the distance to $\partial \hat{I}\text{c◯}$. 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.