The Dirichlet problem for the minimal surface system on smooth domains

TL;DR

This paper establishes the existence of solutions to the Dirichlet problem for the minimal surface system on smooth domains with non-negative Ricci curvature using mean curvature flow, under new small oscillation assumptions.

Contribution

It introduces a novel assumption involving small oscillation and $C^2$ norms, and proves existence without restrictions on domain diameter, including exterior problems.

Findings

Existence of solutions via mean curvature flow.

No restriction on domain diameter.

Application to exterior Dirichlet problems.

Abstract

In this paper, we propose a new assumption (1.2) that involves a small oscillation and $C^2$ norms for maps from smooth bounded domains into Euclidean spaces. Furthermore, by assuming that the domain has non-negative Ricci curvature, we establish the Dirichlet problem for the minimal surface system via the mean curvature flow (MCF) with boundary. The long-time existence of such flow is derived using Bernstein-type theorems of higher codimensional self-shrinkers in the whole space and the half-space. Another novel aspect is that our hypothesis imposes no restriction on the diameter of the domains, which implies an existence result for an exterior Dirichlet problem of the minimal surface system.