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arXiv · 2507.02486

Renormalized variational principles and Hardy-type inequalities

Satyanad Kichenassamy
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Abstract

Let $Ω\subset{\mathbb R}^2$ be a bounded domain on which Hardy's inequality holds. We prove that $[\exp(u^2)-1]/δ^2\in L^1(Ω)$ if $u\in H^1_0(Ω)$, where $δ$ denotes the distance to $\partialΩ$. 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.

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Satyanad Kichenassamy. 2025-07-03. Renormalized variational principles and Hardy-type inequalities. https://arxiv.org/abs/2507.02486

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