A quantitative result for the $k$-Hessian equation
Analysis of PDEs
2025-01-24 v2
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 -Hessian integral, and, with similar arguments, we show a quantitative inequality for the comparison proved by Tso \cite{tso} for solutions to the -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.
Keywords
Cite
@article{arxiv.2407.20811,
title = {A quantitative result for the $k$-Hessian equation},
author = {Alba Lia Masiello and Francesco Salerno},
journal= {arXiv preprint arXiv:2407.20811},
year = {2025}
}