The Dirichlet eigenvalue problems for some concave elliptic Hessian operators

2025/10/19 by Zhang, Jiaogen
#Analysis of PDEs (math.AP) #Differential Geometry (math.DG) #FOS: Mathematics

paper · doi:10.48550/arxiv.2510.16748

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(D²u)=-Λu \textrmin Ω, u=0 \textrmon ∂ 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⊂ ℝⁿ. Furthermore, we prove that the eigenfunction u₁ belongs to C^∞(Ω) ∩ C^1,1(Ω). 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.

Citations

• Global Lipschitz and Sobolev estimates for the Monge-Ampère eigenfunctions of general bounded convex domains
• The Eigenvalue Problem for the Complex Hessian Operator on m-Pseudoconvex Manifolds
• Liouville theorem for a class of Hessian equations
• Determinant majorization and the work of Guo-Phong-Tong and Abja-Olive
• Second order estimates for convex solutions of degenerate k-Hessian equations
• The eigenvalue problem for the Monge-Amp `ere operator on general\n bounded convex domains
• Schauder estimates for degenerate Monge-Ampère equations and smoothness of the eigenfunctions
• Hessian measures II
• The Dirichlet problem for nonlinear second order elliptic equations, III: Functions of the eigenvalues of the Hessian

Cited by

• Fully nonlinear prescribed curvature problems on closed manifolds with negative curvature

Related

• A spectral characterization and an approximation scheme for the Hessian eigenvalue
• The Dirichlet problem for nonlinear second order elliptic equations, III: Functions of the eigenvalues of the Hessian
• Eigenvalue and Dirichlet problem for fully-nonlinear operators in non smooth domains
• Games for eigenvalues of the Hessian and concave/convex envelopes
• Estimates for eigenvalues of a system of of elliptic equations and of the biharmonic operator