Spectral Prescribed Mean Curvature

TL;DR

The paper tackles the computation of surfaces with prescribed mean curvature on 2D domains (rectangles, disks, annuli) under Dirichlet or capillary boundary conditions. It formulates the nonlinear PDE as $\nabla \cdot \frac{\nabla u}{\sqrt{1+|\nabla u|^2}} = \mathcal{H}(u)$ and solves it via an adaptive, modular Newton scheme using spectral discretizations (Chebyshev or Chebyshev–Fourier) and rigorous Fréchet derivatives to linearize the problem at each iteration. Key contributions include explicit operator formulations for multiple geometries, corner handling with measure-theoretic normals, a disk double-cover strategy for stable spectral discretization, and a robust adaptive refinement framework that updates the spectral resolution to meet prescribed tolerances. The numerical results demonstrate successful computation of minimal, constant-mean-curvature, and capillary surfaces on disks and annuli (with some limitations on rectangular domains due to stability), highlighting the method's effectiveness and flexibility. The accompanying open-source Matlab implementations enable practitioners to study prescribed mean curvature problems with diverse boundary data and domain geometries.

Abstract

We consider prescribed mean curvature equations whose solutions are minimal surfaces, constant mean curvature surfaces, or capillary surfaces. We consider both Dirichlet boundary conditions for Plateau problems and nonlinear Neumann boundary conditions for capillary problems and we consider domains in $\mathbf{R}^2$ to be rectangles, disks, or annuli. We present spectral methods for approximating solutions of the associated boundary value problems. These are either based on Chebyshev or Chebyshev-Fourier methods depending on the geometry of the domain. The non-linearity in the prescribed mean curvature equations is treated with a Newton method. The algorithms are designed to be adaptive; if the prescribed tolerances are not met then the resolution of the solution is increased until the tolerances are achieved. 22

Figures (10)

• Figure 1: A minimal surface over an annular domain with a gradient close to blow-up. The left is a plot of the surface height and the right is a contour plot. This provides an illustration of the possible restrictions on the arbitrary boundary data for solutions to exist for non-convex domains. The height of the surface is prescribed to be 0 at the outer radius $b = 2$ and is prescribed to be 1.28792 at the inner radius $a = 1$. If much larger heights are prescribed at $a$ the solution is no longer described by a graph over the base domain, but can be seen to be some catenoid for a further range of heights. While not the only treatment of these results, Bliss Bliss1925 describes the process and the resulting stability analysis nicely.
• Figure 2: A capillary surface over a square domain with a near-optimally extreme contact angle of $\gamma = \pi/4 + 0.035$. The surface height plot on the left shows the gradients getting large, and the contour plot on the right is a helpful companion figure for the proof of blow-up of capillary surfaces as found in ecs.
• Figure 3: The Plateau problem on three different domains. The annular domain is not convex, so solutions do not exist for arbitrary boundary values.
• Figure 4: On the left, the error in the solution of the Plateau problem for a minimal surface on the square with the number of Chebyshev points set to $n=55$. Note that the vertical scale is 1e-12. On the right we plot $2(u_{xx}u_y - u_{xy}u_x)$ which is the coefficient of the $D_x$ term in $F(u)$ for a convergent choice of coefficients given by $g(x,y)= 0.13\sin^2(4\pi x) + 0.1\sin(4\pi y)$.
• Figure 5: The Plateau problem for CMC surfaces on three different domains. The annular domain is not convex, so solutions do not exist for arbitrary boundary values.

Theorems & Definitions (4)

• Theorem 2.1
• Theorem 2.2
• Theorem 2.3
• Theorem 2.4