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Boundary $C^{2, α}$ Regularity for the Oblique Boundary Value Problem of Monge-Ampère Equations

Huaiyu Jian, Xushan Tu

Abstract

We study the good shape property of boundary sections of convex solutions of the oblique boundary value problem for Monge-Ampère equations $$\det D^2u =f(x) \text{ in } Ω, \quad D_βu = φ(x) \text{ on } \partial Ω.$$ In the two-dimensional case, we prove the global $C^{2,α}$ estimate for the solution. When the dimension $n \geq 3$, we show that this estimate still holds if the solution is bounded from above by a quadratic function in the tangent direction. We also obtain an existence result for the convex solution of Monge-Ampère equations with Robin oblique boundary conditions.

Boundary $C^{2, α}$ Regularity for the Oblique Boundary Value Problem of Monge-Ampère Equations

Abstract

We study the good shape property of boundary sections of convex solutions of the oblique boundary value problem for Monge-Ampère equations In the two-dimensional case, we prove the global estimate for the solution. When the dimension , we show that this estimate still holds if the solution is bounded from above by a quadratic function in the tangent direction. We also obtain an existence result for the convex solution of Monge-Ampère equations with Robin oblique boundary conditions.
Paper Structure (13 sections, 40 theorems, 366 equations)

This paper contains 13 sections, 40 theorems, 366 equations.

Table of Contents

  1. Introduction
  2. Preliminaries
  3. A Qualitative Strict Convexity Lemma
  4. Global Lipschitz Regularity of solutions
  5. Good Shape Property
  6. Uniformly Strict Convexity for Normalization Family
  7. The Zero Neumann Boundary Value Problem
  8. Proof of Theorem \ref{['lem:uscl']}
  9. Viscosity Subsolutions and the Compactness of Oblique Boundary Values.
  10. Stationary Lemma and c2 alpha Regularity
  11. Examples
  12. Proof of Lemma \ref{['lem:stationary lem']}
  13. Verification of Example \ref{['exam:c11']}

Key Result

Theorem 1.1

Let $n=2$ and $u \in C(\bar{\Omega})$ be a solution of eq:oblique eq 1. Suppose that $\partial \Omega \in C^{1,\alpha}$, $f \in C^{\alpha}( \bar{\Omega})$, $\phi \in C^{1,\alpha}(\bar{\Omega})$, $\beta \in C^{1,\alpha}(\partial \Omega; \Bbb R^n)$ is an oblique vector field$\beta$ is oblique (poi

Theorems & Definitions (74)

  • Theorem 1.1
  • Theorem 1.2
  • Remark 1.3
  • Theorem 1.4: [JT]
  • Lemma 2.1: John's Lemma
  • Lemma 2.2: [JT]
  • Lemma 2.3
  • Lemma 2.4
  • Lemma 2.5: Aleksandrov’s Maximum Principle
  • Lemma 2.6
  • ...and 64 more