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The Dirichlet problem for Lagrangian mean curvature equation

Arunima Bhattacharya

Abstract

In this paper, we solve the Dirichlet problem with continuous boundary data for the Lagrangian mean curvature equation on a uniformly convex, bounded domain in $\mathbb{R}^n$.

The Dirichlet problem for Lagrangian mean curvature equation

Abstract

In this paper, we solve the Dirichlet problem with continuous boundary data for the Lagrangian mean curvature equation on a uniformly convex, bounded domain in .

Paper Structure

This paper contains 6 sections, 7 theorems, 23 equations.

Table of Contents

  1. Introduction
  2. Preliminaries
  3. $C^{2,\alpha}$ estimate up to the boundary
  4. Proof of Theorem \ref{['main']}
  5. Proof of Theorem \ref{['2.20']}
  6. Appendix

Key Result

Theorem 1.1

Suppose that $\phi\in C^{0}(\partial \Omega)$ and $\psi: \overline \Omega\rightarrow [(n-2)\frac{\pi}{2}+\delta, n\frac{\pi}{2})$ is in $C^{1,1}(\overline \Omega)$, where $\Omega$ is a uniformly convex, bounded domain in $\mathbb{R}^{n}$ and $\delta>0$. Then there exists a unique solution $u\in C^{2

Theorems & Definitions (18)

  • Theorem 1.1
  • Theorem 1.2
  • Remark 1.1
  • Remark 1.2
  • Lemma 2.1
  • proof
  • Theorem 3.1
  • proof
  • Theorem 4.1
  • proof
  • ...and 8 more