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2006 Renormalized variational principles and Hardy-type inequalities
Satyanad Kichenassamy
Differential Integral Equations 19(4): 437-448 (2006).

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.

Citation

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Satyanad Kichenassamy. "Renormalized variational principles and Hardy-type inequalities." Differential Integral Equations 19 (4) 437 - 448, 2006.

Information

Published: 2006
First available in Project Euclid: 21 December 2012

zbMATH: 1212.35122
MathSciNet: MR2215627

Subjects:
Primary: 35J60
Secondary: 35J20 , 42B30

Rights: Copyright © 2006 Khayyam Publishing, Inc.

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Vol.19 • No. 4 • 2006
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