Logo
ScienceStack
Table of Contents
Fetching ...
Sign In

Hölder Regularity of Dirichlet Problem For The Complex Monge-Ampère Equation

Yuxuan Hu, Bin Zhou

Abstract

We study the Dirichlet problem for the complex Monge-Ampère equation on a strictly pseudoconvex domain in Cn or a Hermitian manifold. Under the condition that the right-hand side lies in Lp function and the boundary data are Hölder continuous, we prove the global Hölder continuity of the solution.

Hölder Regularity of Dirichlet Problem For The Complex Monge-Ampère Equation

Abstract

We study the Dirichlet problem for the complex Monge-Ampère equation on a strictly pseudoconvex domain in Cn or a Hermitian manifold. Under the condition that the right-hand side lies in Lp function and the boundary data are Hölder continuous, we prove the global Hölder continuity of the solution.

Paper Structure

This paper contains 9 sections, 10 theorems, 107 equations.

Table of Contents

  1. Introduction
  2. Hölder continuity via regularization
  3. The flat case
  4. The manifold case
  5. Boundary Hölder continuity
  6. Proof of Theorem \ref{['LCMA-dir']}
  7. The flat case in $\mathbb{C}^n$
  8. The manifold case
  9. The case of spaces with isolated singularities

Key Result

Theorem 1.1

Let $(X,\omega)$ be a complete Hermitian manifold, and let $\Omega$ be a relatively compact smooth strictly pseudo-convex open subset of $X$. Suppose $0\leq f\in L^{p}(\Omega,\omega^n)$ for $p>1$ and $\varphi\in C^{\alpha}({\partial\Omega})$. Let $u$ be a solution to the Dirichlet problem: Then for any $0<\gamma,\gamma'<\gamma_n$, $0<\gamma"<\gamma_0$, we have $u\in C^{\alpha'}(\bar{\Omega})$ wit

Theorems & Definitions (19)

  • Theorem 1.1
  • Lemma 2.1
  • proof
  • Remark 2.2
  • Lemma 2.3
  • proof
  • Remark 2.4
  • Theorem 3.1: K2WWZ
  • Lemma 3.2
  • proof
  • ...and 9 more