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Weighted Eigenvalue Problem for a Class of Hessian Equations

Rongxun He, Genggeng Huang

TL;DR

This work tackles the weighted eigenvalue problem for $k$-Hessian equations with weight $|x|^{2sk}$ in strictly $(k-1)$-convex domains containing the origin. The authors develop uniform a priori gradient and Hessian estimates for Hessian equations with the weight, construct eigenpairs via a delta-approximation and compactness, and prove existence, uniqueness (up to scaling), and a variational characterization of the principal eigenvalue. They establish regularity of the eigenfunction in appropriate spaces, including viscosity and weak senses, and derive a weighted embedding that links the eigenvalue to a variational infimum. The results extend classical eigenvalue theory for Monge-Ampère and Hessian equations to a weighted, singular/degenerate setting, with implications for nonlinear potential theory and Hardy-Sobolev-type inequalities.

Abstract

In this paper, we study the existence and uniqueness of solutions to the weighted eigenvalue problem for $k$-Hessian equation. To achieve this, we establish the uniform a priori estimates for gradient and second derivatives of solutions to Hessian equation with weight $|x|^{2sk}$ on the right-hand-side. We also prove that the eigenfunction is a minimizer of the corresponding functional among all $k$-admissible functions vanishing on the boundary.

Weighted Eigenvalue Problem for a Class of Hessian Equations

TL;DR

This work tackles the weighted eigenvalue problem for -Hessian equations with weight in strictly -convex domains containing the origin. The authors develop uniform a priori gradient and Hessian estimates for Hessian equations with the weight, construct eigenpairs via a delta-approximation and compactness, and prove existence, uniqueness (up to scaling), and a variational characterization of the principal eigenvalue. They establish regularity of the eigenfunction in appropriate spaces, including viscosity and weak senses, and derive a weighted embedding that links the eigenvalue to a variational infimum. The results extend classical eigenvalue theory for Monge-Ampère and Hessian equations to a weighted, singular/degenerate setting, with implications for nonlinear potential theory and Hardy-Sobolev-type inequalities.

Abstract

In this paper, we study the existence and uniqueness of solutions to the weighted eigenvalue problem for -Hessian equation. To achieve this, we establish the uniform a priori estimates for gradient and second derivatives of solutions to Hessian equation with weight on the right-hand-side. We also prove that the eigenfunction is a minimizer of the corresponding functional among all -admissible functions vanishing on the boundary.
Paper Structure (6 sections, 17 theorems, 183 equations)

This paper contains 6 sections, 17 theorems, 183 equations.

Table of Contents

  1. Introduction
  2. Preliminaries
  3. Uniform Estimates
  4. Existence Results
  5. Uniqueness Results
  6. Functional Feature

Key Result

Theorem 1.1

Let $\Omega$ be a strictly $(k-1)$-convex bounded domain containing the origin with the boundary $\partial\Omega\in C^{3,1}$. Suppose $s>-s_0$ for $s_0=\min(1,n/2k)$, then there exists a unique positive constant $\lambda_1=\lambda_1(n,k,s,\Omega)$, so that the eigenvalue problem admits a negative solution $\varphi_1\in\Upsilon(\Omega)$ with $\lambda=\lambda_1$, which is unique up to scalar multip

Theorems & Definitions (30)

  • Theorem 1.1
  • Theorem 1.2
  • Proposition 2.1
  • Theorem 3.1
  • proof
  • Theorem 3.2
  • proof
  • Remark 3.1
  • Theorem 3.3
  • proof
  • ...and 20 more