DOI: 10.1093/imrn/rnag237 ISSN: 1073-7928
The Dirichlet problem for the minimal surface system on smooth domains
Caiyan Li, Hengyu ZhouAbstract
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 solve the Dirichlet problem for the minimal surface system (MSS) 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 Euclidean space and in half-spaces. 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 MSS.