The Dirichlet Problem for the Minimal Surface Equation

Definition 1.8.1

The Dirichlet problem for the minimal surface equation asks: given a bounded domain \(\Omega \subset \mathbb{R}^n\) and boundary data \(\phi \in C^0(\partial\Omega)\), find \(u \in C^2(\Omega) \cap C^0(\bar{\Omega})\) such that

\[ \begin{cases} \mathrm{div}\left(\dfrac{\nabla u}{\sqrt{1+|\nabla u|^2}}\right) = 0 & \text{in } \Omega, \\ u = \phi & \text{on } \partial\Omega. \end{cases} \]
Definition 1.8.2

A bounded \(C^2\) domain \(\Omega \subset \mathbb{R}^n\) is mean convex if the mean-curvature vector of \(\partial\Omega\) points weakly into \(\Omega\). Equivalently, with the scalar convention for which Euclidean balls are mean convex, the boundary mean curvature satisfies \(H_{\partial\Omega} \geq 0\).

Theorem 1.8.3

Let \(\Omega \subset \mathbb{R}^n\) be a bounded \(C^2\) domain. Then the Dirichlet problem for the minimal surface equation has a solution \(u \in C^2(\Omega) \cap C^0(\bar{\Omega})\) for every boundary data \(\phi \in C^0(\partial\Omega)\) if and only if \(\Omega\) is mean convex. In this case the solution is unique. If, moreover, \(\partial\Omega\) and \(\phi\) are \(C^{2,\alpha}\), then the solution is \(C^{2,\alpha}\) up to the boundary by the standard boundary regularity theory for quasilinear elliptic equations.

Thus mean convexity is part of the existence theorem, not merely a technical regularity assumption. On a non-mean-convex bounded domain, arbitrary boundary data need not be solvable; Jenkins–Serrin instead prove solvability under an additional smallness condition involving \(\operatorname{osc}(\phi)\) and the first two boundary derivatives of \(\phi\).

Definition 1.8.4

For a minimal graph \(\Sigma_u = \{(x, u(x)) : x \in \Omega\}\), the direction field is defined as

\[ X = \frac{(-\nabla u, 1)}{\sqrt{1 + |\nabla u|^2}}. \]

This direction field satisfies

\[ \begin{aligned} \mathrm{div}\, X &= 0, \\ X \cdot \nu &= 1. \end{aligned} \]

where \(\nu\) is the unit normal to \(\Sigma_u\).

Theorem 1.8.5

Every minimal graph in \(\mathbb{R}^{n+1}\) is area-minimizing within the class of hypersurfaces with the same boundary.

Proof. Define the vector field

\[ V=\frac{(-\nabla u, 1)}{\sqrt{1 + |\nabla u|^2}}. \]

Then the form \(\omega = \iota_V d\mathrm{vol}\) is a calibration, and \(\Sigma_u\) is calibrated by \(\omega\). Therefore, \(\Sigma_u\) minimizes area among all hypersurfaces with the same boundary. ◻

Corollary 1.8.6

Every minimal graph in \(\mathbb{R}^{n+1}\) is stable.

Gaoming Wang
Gaoming Wang
Assistant Professor

My research interests include Geometric Analysis and Partial Differential Equations.