Renormalized variational principles and Hardy-type inequalities

Satyanad Kichenassamy (1)

1 LMR - Laboratoire de Mathématiques de Reims

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.

Domaines: Equations aux dérivées partielles [math.AP]; Analyse fonctionnelle [math.FA]

Satyanad Kichenassamy. Renormalized variational principles and Hardy-type inequalities. Differential and integral equations, 2006, 19, pp.437-448. ⟨hal-00021052⟩