Neumann Problems for Elliptic and Parabolic Sum Hessian Equations

2025/04/07 by Liang, Weizhao, Yan, Jin, Zhu, Hua
Mathematics · #A priori and a posteriori #Analysis of PDEs (math.AP) #Boundary value problem #FOS: Mathematics #Geometric Analysis and Curvature Flows #Geometry and complex manifolds #Hessian equation #Hessian matrix #Neumann boundary condition #Nonlinear Partial Differential Equations #Regular polygon

paper · pdf · doi:10.48550/arxiv.2504.05025

published in arXiv (Cornell University) (Cornell University)

Abstract

This paper studies the Neumann boundary value problem for sum Hessian equations. We first derive a priori C² estimates for (k-1)-admissible solutions in almost convex and uniformly (k-1)-convex domains, and prove the existence of admissible solutions via the method of continuity. Furthermore, we obtain existence results for the classical Neumann problem in uniformly convex domains. Finally, we extend these results to the corresponding parabolic problems.

Citations

• The Pogorelov estimates for the sum Hessian equation with rigidity theorem and parabolic versions
• The Neumann problem of special Lagrangian type equations
• Pogorelov type C² estimates for Sum Hessian equations and a rigidity theorem
• Gradient estimate of the solutions to Hessian equations with oblique boundary value
• The Neumann problem of Hessian quotient equations
• The Neumann problem for a class of mixed complex Hessian equations
• The classical Neumann problem for a class of mixed Hessian equations
• The Neumann Problem for Parabolic Hessian Quotient Equations
• A class of curvature type equations
• The curvature estimates for convex solutions of some fully nonlinear Hessian type equations
• Classical Neumann Problems for Hessian equations and Alexandrov-Fenchel's inequalities
• Oblique boundary value problems for augmented Hessian equations III
• Global C² estimates for convex solutions of curvature equations
• Global C2‐Estimates for Convex Solutions of Curvature Equations
• Hypersurfaces of prescribed curvature measure
• Hypersurfaces in Hⁿ⁺¹ and conformally invariant equations: the generalized Christoffel and Nirenberg problems
• Hessian equations with elementary symmetric functions
• On Harnack inequalities and singularities of admissible metrics in the Yamabe problem
• On Harnack inequalities and singularities of admissible metrics in the Yamabe problem
• Translating solutions for Gauss curvature flows with Neumann boundary conditions
• The Dirichlet problem for nonlinear second order elliptic equations, III: Functions of the eigenvalues of the Hessian
• The dirichlet problem for nonlinear second‐order elliptic equations I. Monge‐ampégre equation
• Calibrated geometries

Related

• The Neumann Problem for Parabolic Hessian Quotient Equations
• Classical Neumann Problems for Hessian equations and Alexandrov-Fenchel's inequalities
• The classical Neumann problem for a class of mixed Hessian equations
• Pogorelov type C² estimates for Sum Hessian equations and a rigidity theorem
• A Serrin-type over-determined problem for Hessian equations in the exterior domain