Renormalized variational principles and Hardy-type inequalities
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arXiv
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| Format: | Preprint |
| Veröffentlicht: |
2025
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| _version_ | 1866911036544647168 |
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| author | Kichenassamy, Satyanad |
| author_facet | Kichenassamy, Satyanad |
| contents | 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. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2507_02486 |
| institution | arXiv |
| publishDate | 2025 |
| record_format | arxiv |
| spellingShingle | Renormalized variational principles and Hardy-type inequalities Kichenassamy, Satyanad Analysis of PDEs Functional Analysis 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. |
| title | Renormalized variational principles and Hardy-type inequalities |
| topic | Analysis of PDEs Functional Analysis |
| url | https://arxiv.org/abs/2507.02486 |