Existence and non-existence of minimal graphs

TL;DR

This paper investigates the existence and non-existence of minimal graphs over mean convex domains using mean curvature flow, extending classical results and identifying conditions for solvability and non-solvability of the Dirichlet problem.

Contribution

It generalizes the classical Jenkins-Serrin criterion to higher dimensions and codimensions, and constructs examples where the Dirichlet problem fails to have solutions.

Findings

Existence of minimal graphs over mean convex domains for a broad class of boundary data.

Construction of boundary data on mean convex domains where the Dirichlet problem is not solvable in codimension 2.

Analysis of existence and uniqueness of minimal graphs via perturbation methods.

Abstract

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 $C^2$ domains for a large class of prescribed 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.