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 σ$$(Σ, g) L(∂Σ, g) over a class of smooth metrics, g, where σ$$(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.
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Keywords: Steklov eigenvalue, eigenvalue optimization, free boundary minimal surface
@article{COCV_2021__27_1_A36_0,
author = {Oudet, \'Edouard and Kao, Chiu-Yen and Osting, Braxton},
title = {Computation of free boundary minimal surfaces \protect\emph{via} extremal {Steklov} eigenvalue problems},
journal = {ESAIM: Control, Optimisation and Calculus of Variations},
year = {2021},
publisher = {EDP-Sciences},
volume = {27},
doi = {10.1051/cocv/2021033},
language = {en},
url = {https://www.numdam.org/articles/10.1051/cocv/2021033/}
}
TY - JOUR AU - Oudet, Édouard AU - Kao, Chiu-Yen AU - Osting, Braxton TI - Computation of free boundary minimal surfaces via extremal Steklov eigenvalue problems JO - ESAIM: Control, Optimisation and Calculus of Variations PY - 2021 VL - 27 PB - EDP-Sciences UR - https://www.numdam.org/articles/10.1051/cocv/2021033/ DO - 10.1051/cocv/2021033 LA - en ID - COCV_2021__27_1_A36_0 ER -
%0 Journal Article %A Oudet, Édouard %A Kao, Chiu-Yen %A Osting, Braxton %T Computation of free boundary minimal surfaces via extremal Steklov eigenvalue problems %J ESAIM: Control, Optimisation and Calculus of Variations %D 2021 %V 27 %I EDP-Sciences %U https://www.numdam.org/articles/10.1051/cocv/2021033/ %R 10.1051/cocv/2021033 %G en %F COCV_2021__27_1_A36_0
Oudet, Édouard; Kao, Chiu-Yen; Osting, Braxton. Computation of free boundary minimal surfaces via extremal Steklov eigenvalue problems. ESAIM: Control, Optimisation and Calculus of Variations, Tome 27 (2021), article no. 34. doi: 10.1051/cocv/2021033
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