Existence and non-existence of minimal graphs

We study the Dirichlet problem for minimal surface systems in arbitrary dimension and codimension via mean curvature flow, and obtain the existence of minimal graphs over arbitrary mean convex bounded C2 domains for a large class of pre-scribed boundary data. This result can be seen as a natural generalization of the classical sharp criterion for solvability of the minimal surface equation by Jenkins-Serrin. In contrast, we also construct a class of prescribed boundary data on just mean convex domains for which the Dirichlet problem in codimension 2 is not solvable. Moreover, we study existence and the uniqueness of minimal graphs by perturbation.(c) 2023 Elsevier Masson SAS. All rights reserved.

Automated summary

This study tackles the Dirichlet problem for minimal surface systems in various dimensions and codimensions through mean curvature flow, establishing the existence of minimal graphs over arbitrary mean convex bounded C2 domains with certain prescribed boundary data. It serves as a natural extension of the classical sharp criterion for solvability of the minimal surface equation by Jenkins-Serrin. However, a class of prescribed boundary data on mean convex domains was identified where the Dirichlet problem in codimension 2 remains unsolvable. Additionally, the study delves into the existence and uniqueness of minimal graphs through perturbation.