An alternative solvability criterion for the Dirichlet problem for the minimal surface equation and an application to the mean curvature flow

1. Problem Statement

The paper addresses the Dirichlet problem for the Minimal Surface Equation (MSE) on a complete $n$-dimensional Riemannian manifold $M$ ($n \geq 2$). The goal is to find a function $u \in C^2(\Omega)$ such that:
$$\begin{cases} M(u) = (1 + |\nabla u|^2)\Delta u - \nabla^2 u(\nabla u, \nabla u) = 0 & \text{in } \Omega, \\ u = \phi & \text{on } \partial\Omega, \end{cases}$$
where $\Omega \subset M$ is a $C^2$ domain and $\phi \in C^2(\Omega)$ is the boundary data.

The Core Difficulty:
Classical solvability results (e.g., Jenkins–Serrin theory) rely heavily on the domain being mean convex (i.e., the mean curvature of the boundary $H_{\partial\Omega} \geq 0$). When the domain is not mean convex (specifically, when the set $\partial\Omega^- = \{p \in \partial\Omega : H_{\partial\Omega}(p) < 0\}$ is non-empty), solvability is not guaranteed for arbitrary boundary data.
Existing criteria for non-mean convex domains (such as those by Jenkins–Serrin and subsequent generalizations) require the oscillation of the boundary data $\phi$ to be bounded by a constant $B$. However, calculating $B$ is notoriously difficult because:

1. It depends on local geometric properties of the boundary in a complex, non-explicit way.
2. It couples the geometry of the domain and the derivatives of the boundary function $\phi$, making it hard to treat them independently.
3. It is difficult to apply to unbounded domains.

2. Methodology

The authors propose a novel approach based on the construction of explicit boundary barriers derived from a second-order Ordinary Differential Equation (ODE).

A. Geometric Setup

• Exterior Sphere Condition (r-es): The authors define a condition where, at every point $p$ in the non-mean convex part of the boundary ($\partial\Omega^-$), there exists an exterior geodesic sphere of radius $r$ tangent to the boundary.
• Locally Hadamard r-es Condition: A refinement for Hadamard manifolds (non-positive curvature) ensuring that the mean curvature of the exterior spheres behaves monotonically as one moves away from the boundary.
• Key Geometric Parameters:
  • $\lambda_r$: The infimum of the mean curvature of the exterior spheres at the contact points (negative value).
  • $\mu_r$: The supremum of the norm of the Hessian of the distance function to these spheres.

B. Barrier Construction

Instead of using complex geometric bounds, the authors construct barrier functions of the form:
$$w(x) = \phi(x) \pm \psi(d(x))$$
where $d(x)$ is the distance to the boundary (or exterior spheres) and $\psi$ is a scalar function solving a specific ODE.

1. Derivation of the ODE: By analyzing the minimal surface operator $M(w)$ and estimating terms involving the derivatives of $\phi$ and the geometry of the domain, the authors derive a differential inequality.
2. The Governing ODE: This inequality is satisfied if $\psi$ solves:
   $$\psi'' - \sigma \psi' + \varrho = 0$$
   where $\sigma$ and $\varrho$ are constants determined by the $C^2$-norm of $\phi$ (denoted $\tau$) and the dimension $n$.
3. Explicit Solution: The authors solve this ODE explicitly to obtain $\psi(s)$, which serves as the "height" of the barrier. The existence of a valid barrier depends on the oscillation of $\phi$ ($\omega_\phi$) satisfying a specific inequality involving $\lambda_r, \mu_r$, and the coefficients of the ODE.

3. Key Contributions

1. An Alternative Solvability Criterion

The paper establishes a new sufficient condition for the existence of solutions to the Dirichlet problem when $\partial\Omega^-$ is non-empty.

• Independence: The condition separates geometric constraints (domain shape) from analytic constraints (boundary data). The oscillation bound is given by an explicit formula involving $\lambda_r$ and $\mu_r$, rather than a complex neighborhood-dependent constant $B$.
• Applicability: The criterion applies to unbounded domains and Hadamard manifolds, extending beyond the classical bounded Euclidean setting.
• Explicitness: The condition is computationally verifiable. For $\mathbb{R}^n$, it reduces to a simple inequality relating the oscillation of $\phi$, the radius $r$ of the exterior sphere, and the derivatives of $\phi$.

2. Application to Mean Curvature Flow (MCF)

The authors extend their elliptic barrier construction to the parabolic setting (Graphical Mean Curvature Flow):
$$\partial_t u = M(u)$$

• Short-Time Existence: They prove that if the structural conditions for the elliptic problem are met, the parabolic flow admits a short-time classical solution $u \in C^{2,1}$.
• Preservation of Graphicality: Crucially, the flow remains a graph over $\Omega$ for a time $T > 0$, even if the boundary $\partial\Omega$ is not mean convex.
• Mechanism: The same ODE-derived function $\psi$ acts as a parabolic barrier, preventing the gradient $|\nabla u|$ from blowing up (which would cause the graph to lose its graphical nature).

4. Main Results

Theorem 3 (Elliptic Solvability)

Let $\Omega \subset M$ be a $C^2$ domain. If $\partial\Omega^-$ satisfies the locally Hadamard r-es condition, and the oscillation of the boundary data $\omega_\phi$ satisfies:
$$\omega_\phi \leq \frac{1}{\varrho(\theta-1)^2} \left[ \theta(1 - e^{-\varrho(\theta-1)\delta}) - \varrho(\theta-1)\delta \right]$$
(where $\theta, \varrho, \delta$ are defined via the geometry and $\tau(\phi)$), then the Dirichlet problem admits a unique solution (if $\Omega$ is bounded).

Corollary 4 (Hadamard Manifolds)

For Hadamard manifolds, the condition simplifies to an explicit inequality relating the domain geometry ($\lambda_r$) and the boundary data oscillation ($\omega_\phi$):
$$\frac{\sqrt{a} - \ln a - 1 - \sqrt{\varrho}\omega(a + c\mu_r - 1)}{\sqrt{\varrho}\omega b(n-1)} \leq \lambda_r$$
This allows for a direct check of solvability without computing complex neighborhood constants.

Theorem 10 (Parabolic Existence)

Under the same structural conditions, the initial-boundary value problem for the graphical mean curvature flow admits a unique classical solution for a short time $T > 0$. The solution remains a graph over $\Omega$ throughout this interval, regardless of the lack of mean convexity of the boundary.

5. Significance and Impact

1. Overcoming Classical Limitations: The paper provides a tool to solve the MSE and MCF in domains where the classical Jenkins–Serrin theory fails. The authors demonstrate this with a specific example in $\mathbb{R}^2$ where the Jenkins–Serrin bound $B$ is too small to allow the given boundary data, yet their new criterion guarantees a solution.
2. Decoupling Geometry and Data: By making the geometric restrictions and boundary data enter the solvability condition independently, the method offers greater flexibility and clarity for researchers analyzing specific domains or data sets.
3. Bridge Between Elliptic and Parabolic: The work elegantly connects the static (elliptic) and dynamic (parabolic) theories. The same ODE-based barrier construction that ensures the existence of a minimal surface also ensures the short-time existence and regularity of the mean curvature flow.
4. Removal of Mean Convexity Assumption: A major theoretical hurdle in geometric analysis is the requirement of mean convexity for boundary regularity. This paper shows that for graphs, this assumption can be replaced by a structural condition on the boundary data and the "exterior sphere" geometry, significantly broadening the scope of solvable problems.

In summary, the paper introduces a robust, ODE-based framework for constructing barriers that simplifies the verification of solvability for minimal surfaces and mean curvature flows in non-convex and unbounded settings, offering a practical alternative to classical, often intractable, geometric criteria.