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Neumann Problems for Elliptic and Parabolic Sum Hessian Equations

Weizhao Liang, Jin Yan, Hua Zhu

TL;DR

This work advances the solvability theory for Neumann problems of sum Hessian equations by establishing sharp a priori $C^2$ estimates in almost convex and uniformly $(k-1)$-convex domains, enabling existence results via the continuity method for both the elliptic and parabolic problems. It introduces the notion of $(k-1)$-admissible solutions and leverages Garding-type inequalities and boundary barriers to control double-normal derivatives, extending the theory to the parabolic setting with long-time behavior and convergence to stationary translating solutions. The paper also treats the classical Neumann problem through a perturbation-perturbation limit and provides parallel results for the parabolic augmented Hessian equation, including exponential convergence under favorable data conditions. Overall, the results broaden solvability for augmented Hessian Neumann problems in non-strictly convex domains and furnish a robust framework for parabolic analogues and asymptotic analysis.

Abstract

This paper studies the Neumann boundary value problem for sum Hessian equations. We first derive a priori $C^2$ 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.

Neumann Problems for Elliptic and Parabolic Sum Hessian Equations

TL;DR

This work advances the solvability theory for Neumann problems of sum Hessian equations by establishing sharp a priori estimates in almost convex and uniformly -convex domains, enabling existence results via the continuity method for both the elliptic and parabolic problems. It introduces the notion of -admissible solutions and leverages Garding-type inequalities and boundary barriers to control double-normal derivatives, extending the theory to the parabolic setting with long-time behavior and convergence to stationary translating solutions. The paper also treats the classical Neumann problem through a perturbation-perturbation limit and provides parallel results for the parabolic augmented Hessian equation, including exponential convergence under favorable data conditions. Overall, the results broaden solvability for augmented Hessian Neumann problems in non-strictly convex domains and furnish a robust framework for parabolic analogues and asymptotic analysis.

Abstract

This paper studies the Neumann boundary value problem for sum Hessian equations. We first derive a priori estimates for -admissible solutions in almost convex and uniformly -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.

Paper Structure

This paper contains 20 sections, 28 theorems, 231 equations.

Table of Contents

  1. Introduction
  2. Preliminaries
  3. $C^0$ and $C^1$ Estimates
  4. $C^0$ estimate
  5. Gradient estimate
  6. Second-Order Derivatives Estimate
  7. Reduce global second estimates to boundary double normal derivatives
  8. An upper estimate of the double normal derivatives on the boundary
  9. A lower estimate of the double normal derivatives on the boundary
  10. Existence of the Boundary Problems
  11. Proof of Theorem \ref{['thm 1.2-Neumann for elliptic sum']}
  12. Proof of Theorem \ref{['thm 1.3-classical Neumann for elliptic sum']}
  13. Neumann problem for parabolic sum Hessian equation
  14. $u_t$ and $C^0$ estimates
  15. Gradient estimate
  16. ...and 5 more sections

Key Result

Theorem 1.2

Let $\Omega \subset \mathbb{R}^n$ be a $C^4$ almost convex and uniformly $(k-1)$-convex domain, $f \in C^{2}(\overline{\Omega})$, $\varphi \in C^{3}(\partial \Omega\times\mathbb{R})$ satisfy $f>0$, $\varphi_u\leq c_\varphi<0$, and $2a_{\kappa}>c_{\varphi}$. Then there exists a unique $(k-1)$-admissi where $\gamma\in (0,1)$ is a constant.

Theorems & Definitions (54)

  • Definition 1.1
  • Theorem 1.2
  • Remark 1.3
  • Theorem 1.4
  • Remark 1.5
  • Theorem 1.6
  • Theorem 1.7
  • Definition 2.1
  • Lemma 2.2
  • proof
  • ...and 44 more