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A quantitative result for the $k$-Hessian equation

Alba Lia Masiello, Francesco Salerno

TL;DR

The paper develops a quantitative Polya-Szego-type theory for the $k$-Hessian equation by introducing a $(k-1)$-symmetrization that preserves quermassintegrals of sublevel sets. Leveraging the quantitative Alexandrov-Fenchel inequality and the Hausdorff asymmetry, it proves explicit deficit bounds for the $k$-Hessian functional under shape perturbations and derives a sharp comparison between the symmetrized and radially symmetric solutions. These results yield quantitative versions of Faber-Krahn and Saint-Venant-type inequalities, as well as eigenvalue and torsional rigidity stability statements in terms of the asymmetry of the domain. The approach unifies convex geometry with fully nonlinear PDE techniques and provides stability estimates for shape optimization problems involving the $k$-Hessian operator in high dimensions.

Abstract

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.

A quantitative result for the $k$-Hessian equation

TL;DR

The paper develops a quantitative Polya-Szego-type theory for the -Hessian equation by introducing a -symmetrization that preserves quermassintegrals of sublevel sets. Leveraging the quantitative Alexandrov-Fenchel inequality and the Hausdorff asymmetry, it proves explicit deficit bounds for the -Hessian functional under shape perturbations and derives a sharp comparison between the symmetrized and radially symmetric solutions. These results yield quantitative versions of Faber-Krahn and Saint-Venant-type inequalities, as well as eigenvalue and torsional rigidity stability statements in terms of the asymmetry of the domain. The approach unifies convex geometry with fully nonlinear PDE techniques and provides stability estimates for shape optimization problems involving the -Hessian operator in high dimensions.

Abstract

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 -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.
Paper Structure (9 sections, 9 theorems, 141 equations, 1 figure)

This paper contains 9 sections, 9 theorems, 141 equations, 1 figure.

Table of Contents

  1. Introduction
  2. Notation and preliminaries
  3. Quermassintegrals
  4. Symmetrization
  5. The k-Hessian operator
  6. Proofs of main results.
  7. Proof of Theorem \ref{['theorem1.1']}
  8. Proof of Theorem \ref{['teo1.2']}
  9. Proof of Theorems \ref{['gradienti']} and \ref{['teo1.3']}

Key Result

Theorem 1.1

Let $\Omega$ be an open, bounded, and convex set of $\mathbb{R}^n$ and let $u\in C^2(\Omega)$ be a convex function that vanishes on the boundary of $\Omega$. Then, there exists a positive constant $C_1=C_1(n,k,\Omega)$ such that where $k=1,\dots,n-1$, $\beta_n$ is given in Theorem quantitative:af, and $\zeta_{k-1}(\Omega)$ is the $(k-1)$-th meanradius of $\Omega$.

Figures (1)

  • Figure 1: Control of the quermassintegrals of the sublevelsets of $u$ via the quermassintegrals of $\Omega$.

Theorems & Definitions (34)

  • Theorem 1.1
  • Theorem 1.2
  • Theorem 1.3
  • Theorem 1.4
  • Definition 2.1
  • Definition 2.2
  • Definition 2.3
  • Definition 2.4
  • Definition 2.5
  • Remark 2.1
  • ...and 24 more