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The Dirichlet eigenvalue problems for some concave elliptic Hessian operators

Jiaogen Zhang

TL;DR

This work analyzes the Dirichlet eigenvalue problem for a broad class of concave elliptic Hessian operators $F(D^2u)$ on smooth, strictly $\Gamma$-convex domains, encompassing Monge–Ampère, $k$-Hessian, and $p$-Monge–Ampère operators. Under Condition (T), the authors establish the existence and regularity of the first $\Gamma$-admissible eigenpair $(u_1,\Lambda_1)$, with $u_1\in C^{\infty}(\Omega)\cap C^{1,1}(\overline{\Omega})$, and show uniqueness up to a multiplicative constant. They develop comprehensive a priori estimates, including global gradient and Hessian bounds, and implement Lions’ approximation to realize the eigenpair; they further extend the framework to invariant Gårding–Dirichlet operators. A bifurcation-type theory is developed around the first eigenvalue, linking the eigenproblem to a nonlinear Dirichlet problem and providing a continuum of solutions via a continuity method. Overall, the results advance fully nonlinear spectral theory, connect to convex geometry, and illuminate bifurcation phenomena for a wide class of Hessian-type operators.

Abstract

In this manuscript, we investigate a priori estimates for the solution to the Dirichlet eigenvalue problem for a broad class of concave elliptic Hessian operators of the form \[ F(D^2u)=-Λu \quad \textrm{in} \, Ω, \qquad u=0 \quad \textrm{on} \, \partial Ω. \] These operators encompass the Monge-Ampère operator, the $k$-Hessian operators, and the $p$-Monge-Ampère operators. We impose a fairly mild constraint on the operator $F$, allowing us to demonstrate the existence of the first nonzero eigenvalue and its corresponding $Γ$-admissible eigenfunction on the smooth, strictly $Γ$-convex domain $Ω\subset \mathbb{R}^{n}$. Furthermore, we prove that the eigenfunction $u_{1}$ belongs to $C^{\infty}(Ω) \cap C^{1,1}(\overlineΩ)$. As an application, we prove that every invariant Gårding-Dirichlet operator admits a unique first nonzero eigenvalue. Finally, a bifurcation-type theory for these operators is also established.

The Dirichlet eigenvalue problems for some concave elliptic Hessian operators

TL;DR

This work analyzes the Dirichlet eigenvalue problem for a broad class of concave elliptic Hessian operators on smooth, strictly -convex domains, encompassing Monge–Ampère, -Hessian, and -Monge–Ampère operators. Under Condition (T), the authors establish the existence and regularity of the first -admissible eigenpair , with , and show uniqueness up to a multiplicative constant. They develop comprehensive a priori estimates, including global gradient and Hessian bounds, and implement Lions’ approximation to realize the eigenpair; they further extend the framework to invariant Gårding–Dirichlet operators. A bifurcation-type theory is developed around the first eigenvalue, linking the eigenproblem to a nonlinear Dirichlet problem and providing a continuum of solutions via a continuity method. Overall, the results advance fully nonlinear spectral theory, connect to convex geometry, and illuminate bifurcation phenomena for a wide class of Hessian-type operators.

Abstract

In this manuscript, we investigate a priori estimates for the solution to the Dirichlet eigenvalue problem for a broad class of concave elliptic Hessian operators of the form These operators encompass the Monge-Ampère operator, the -Hessian operators, and the -Monge-Ampère operators. We impose a fairly mild constraint on the operator , allowing us to demonstrate the existence of the first nonzero eigenvalue and its corresponding -admissible eigenfunction on the smooth, strictly -convex domain . Furthermore, we prove that the eigenfunction belongs to . As an application, we prove that every invariant Gårding-Dirichlet operator admits a unique first nonzero eigenvalue. Finally, a bifurcation-type theory for these operators is also established.
Paper Structure (15 sections, 16 theorems, 162 equations)

This paper contains 15 sections, 16 theorems, 162 equations.

Table of Contents

  1. Introduction
  2. Preliminaries
  3. Convex invariant cones and concave elliptic Hessian operators
  4. First and second derivatives of $F$
  5. First order estimates
  6. Second order estimates
  7. Global second order estimate
  8. Boundary second order estimate
  9. Double tangential estimates
  10. Tangential to normal estimates
  11. Double normal estimates
  12. The Dirichlet eigenvalue problem
  13. Lions' approximation strategy
  14. Proof of Theorem \ref{['eigenvalue theorem']}
  15. Application to the bifurcation-type theory

Key Result

Theorem 1.2

Let $\Omega$ be a smooth bounded and strictly $\Gamma$-convex domain within $\mathbb{R}^n$. Assume $F$ is a concave elliptic Hessian operator on an open convex cone $\mathscr{M}(\Gamma)\subset \mathrm{Sym}^2(\mathbb{R}^n)$ that satisfies the Condition (T). Then, there exists a positive number $\Lam

Theorems & Definitions (33)

  • Definition 1.1
  • Theorem 1.2
  • Remark 1.3
  • Corollary 1.4
  • Theorem 1.5
  • Definition 2.1
  • Definition 2.2
  • Definition 2.3
  • Lemma 2.4
  • proof
  • ...and 23 more