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arXiv · 2407.20811

A quantitative result for the $k$-Hessian equation

Alba Lia Masiello·Francesco Salerno
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Abstract

In this paper, we study a symmetrization that preserves the mixed volume of the sublevel sets of a convex function, under which, a P\'olya-Szeg\H o type inequality holds. We refine this symmetrization to obtain a quantitative improvement of the P\'olya-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.

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Alba Lia Masiello, Francesco Salerno. 2024-07-30. A quantitative result for the $k$-Hessian equation. https://arxiv.org/abs/2407.20811

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