The Dirichlet problem for Lagrangian mean curvature equation

TL;DR

This paper addresses solving the Dirichlet problem for the Lagrangian mean curvature equation with continuous boundary data on convex domains, advancing understanding of geometric PDEs in differential geometry.

Contribution

It provides existence results for the Dirichlet problem for the Lagrangian mean curvature equation on convex domains, a problem previously unresolved for continuous boundary data.

Findings

Established existence of solutions for the Dirichlet problem

Extended solvability to continuous boundary data

Focused on uniformly convex bounded domains

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 $\mathbb{R}^n$.