Computation of free boundary minimal surfaces via extremal Steklov eigenvalue problems

Author:

Oudet ÉdouardORCID,Kao Chiu-Yen,Osting Braxton

Abstract

Recently Fraser and Schoen showed that the solution of a certain extremal Steklov eigenvalue problem on a compact surface with boundary can be used to generate a free boundary minimal surface, i.e., a surface contained in the ball that has (i) zero mean curvature and (ii) meets the boundary of the ball orthogonally (doi:10.1007/s00222-015-0604-x). In this paper, we develop numerical methods that use this connection to realize free boundary minimal surfaces. Namely, on a compact surface, Σ, with genus γ and b boundary components, we maximize σj(Σ, gL(Σ, g) over a class of smooth metrics, g, where σj(Σ, g) is the jth nonzero Steklov eigenvalue and L(Σ, g) is the length of Σ. Our numerical method involves (i) using conformal uniformization of multiply connected domains to avoid explicit parameterization for the class of metrics, (ii) accurately solving a boundary-weighted Steklov eigenvalue problem in multi-connected domains, and (iii) developing gradient-based optimization methods for this non-smooth eigenvalue optimization problem. For genus γ = 0 and b = 2, …, 9, 12, 15, 20 boundary components, we numerically solve the extremal Steklov problem for the first eigenvalue. The corresponding eigenfunctions generate a free boundary minimal surface, which we display in striking images. For higher eigenvalues, numerical evidence suggests that the maximizers are degenerate, but we compute local maximizers for the second and third eigenvalues with b = 2 boundary components and for the third and fifth eigenvalues with b = 3 boundary components.

Publisher

EDP Sciences

Subject

Computational Mathematics,Control and Optimization,Control and Systems Engineering

Cited by 7 articles. 订阅此论文施引文献 订阅此论文施引文献,注册后可以免费订阅5篇论文的施引文献,订阅后可以查看论文全部施引文献

1. From Steklov to Laplace: free boundary minimal surfaces with many boundary components;Duke Mathematical Journal;2024-06-01

2. Shape optimization for combinations of Steklov eigenvalues on Riemannian surfaces;Mathematische Zeitschrift;2024-04-15

3. Harmonic Functions on Finitely Connected Tori;SIAM Journal on Numerical Analysis;2023-11-20

4. Some recent developments on the Steklov eigenvalue problem;Revista Matemática Complutense;2023-09-28

5. Extremal Graph Realizations and Graph Laplacian Eigenvalues;SIAM Journal on Discrete Mathematics;2023-07-28

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