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The Neumann problem for a class of Hessian quotient type equations

Jiabao Gong, Zixuan Liu, Qiang Tu

TL;DR

This work addresses the Neumann problem for a Hessian quotient type equation in a Euclidean domain, where the right-hand side depends on $(x,u,Du)$. The authors derive interior gradient estimates under a growth condition, then establish global gradient and second-order estimates by reducing the problem to boundary double normal derivatives and performing a meticulous boundary analysis. The combination of these a priori estimates with Evans–Krylov theory and the method of continuity yields existence and uniqueness of a $(\Lambda,k)$-convex solution in $C^{2,\alpha}(\overline{\Omega})$ to the Neumann problem, along with corollaries for specific right-hand side forms. Overall, the paper extends the Neumann theory for Hessian quotient equations, providing a robust framework for interior and boundary regularity and a priori control in this nonlinear setting.

Abstract

In this paper, we consider the Neumann problem for a class of Hessian quotient equations involving a gradient term on the right-hand side in Euclidean space. More precisely, we derive the interior gradient estimates for the $(Λ, k)$-convex solution of Hessian quotient equation $\frac{σ_k(Λ(D^2 u))}{σ_l(Λ(D^2 u))}=ψ(x,u,D u)$ with $0\leq l<k\leq C^{p-1}_{n-1}$ under the assumption of the growth condition. As an application, we obtain the global a priori estimates and the existence theorem for the Neumann problem of this Hessian quotient type equation.

The Neumann problem for a class of Hessian quotient type equations

TL;DR

This work addresses the Neumann problem for a Hessian quotient type equation in a Euclidean domain, where the right-hand side depends on . The authors derive interior gradient estimates under a growth condition, then establish global gradient and second-order estimates by reducing the problem to boundary double normal derivatives and performing a meticulous boundary analysis. The combination of these a priori estimates with Evans–Krylov theory and the method of continuity yields existence and uniqueness of a -convex solution in to the Neumann problem, along with corollaries for specific right-hand side forms. Overall, the paper extends the Neumann theory for Hessian quotient equations, providing a robust framework for interior and boundary regularity and a priori control in this nonlinear setting.

Abstract

In this paper, we consider the Neumann problem for a class of Hessian quotient equations involving a gradient term on the right-hand side in Euclidean space. More precisely, we derive the interior gradient estimates for the -convex solution of Hessian quotient equation with under the assumption of the growth condition. As an application, we obtain the global a priori estimates and the existence theorem for the Neumann problem of this Hessian quotient type equation.
Paper Structure (10 sections, 11 theorems, 117 equations)

This paper contains 10 sections, 11 theorems, 117 equations.

Table of Contents

  1. Introduction
  2. Preliminaries
  3. $C^0$ estimates
  4. global gradient estimates
  5. Internal gradient estimates
  6. Boundary gradient estimates
  7. global estimates for second-order derivatives
  8. Reduce global second-order derivatives to double normal second-order derivatives on the boundary
  9. Double normal estimates on the boundary
  10. Proof of the main theorem

Key Result

Theorem 1.2

Let $0\leq l<k\leq C^{p-1}_{n-1}$ and $u\in C^3(B_r(0))$ be the $(\Lambda,k)$-convex solution of equation (1.1) in $B_r(0)$. Suppose that $\tilde{\psi}\in C^1(B_r\times\mathbb{R}\times\mathbb{R}^n)$ with $\tilde{\psi}>0$ satisfy the Condition growth. Then where $C$ is a positive constant depending on $n, k, l, p, M_1$ and $C_1$.

Theorems & Definitions (22)

  • Definition 1.1
  • Theorem 1.2
  • Remark 1.3
  • Theorem 1.4
  • Corollary 1.5
  • Proposition 2.1
  • Proposition 2.2
  • Proposition 2.3
  • proof
  • Theorem 3.1
  • ...and 12 more