A quantitative result for the $k$-Hessian equation
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arXiv
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| Format: | Preprint |
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2024
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| _version_ | 1866915118121484288 |
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| author | Masiello, Alba Lia Salerno, Francesco |
| author_facet | Masiello, Alba Lia Salerno, Francesco |
| contents | In this paper, we study a symmetrization that preserves the mixed volume of the sublevel sets of a convex function, under which, a Pólya-Szeg\H o type inequality holds. We refine this symmetrization to obtain a quantitative improvement of the Pólya-Szeg\H o inequality for the $k$-Hessian integral, and, with similar arguments, we show a quantitative inequality for the comparison proved by Tso \cite{tso} for solutions to the $k$-Hessian equation.
As an application of the first result, we prove a quantitative version of the Faber-Krahn and Saint-Venant inequalities for these equations. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2407_20811 |
| institution | arXiv |
| publishDate | 2024 |
| record_format | arxiv |
| spellingShingle | A quantitative result for the $k$-Hessian equation Masiello, Alba Lia Salerno, Francesco Analysis of PDEs 52A39, 35B35, 35J60, 35J96 In this paper, we study a symmetrization that preserves the mixed volume of the sublevel sets of a convex function, under which, a Pólya-Szeg\H o type inequality holds. We refine this symmetrization to obtain a quantitative improvement of the Pólya-Szeg\H o inequality for the $k$-Hessian integral, and, with similar arguments, we show a quantitative inequality for the comparison proved by Tso \cite{tso} for solutions to the $k$-Hessian equation. As an application of the first result, we prove a quantitative version of the Faber-Krahn and Saint-Venant inequalities for these equations. |
| title | A quantitative result for the $k$-Hessian equation |
| topic | Analysis of PDEs 52A39, 35B35, 35J60, 35J96 |
| url | https://arxiv.org/abs/2407.20811 |