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1. Computation of free boundary minimal surfaces via extremal Steklov eigenvalue problems

   https://doi.org/10.1051/cocv/2021033  

   Oudet, Édouard; Kao, Chiu-Yen; Osting, Braxton (January 2021, ESAIM: Control, Optimisation and Calculus of Variations)

   null (Ed.)

   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 j (Σ, g )  L ( ∂ Σ, g ) over a class of smooth metrics, g , where σ j (Σ, g ) is the j th 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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2. Computational approaches for extremal geometric eigenvalue problems

   C.-Y. Kao; B. Osting; and E ́. Oudet (January 2023, Handbook of numerical analysis)

   In an extremal eigenvalue problem, one considers a family of eigenvalue problems, each with discrete spectra, and extremizes a chosen eigenvalue over the family. In this chapter, we consider eigenvalue problems defined on Riemannian manifolds and extremize over the metric structure. For example, we consider the problem of maximizing the principal Laplace–Beltrami eigenvalue over a family of closed surfaces of fixed volume. Computational approaches to such extremal geometric eigenvalue problems present new computational challenges and require novel numerical tools, such as the parameterization of conformal classes and the development of accurate and efficient methods to solve eigenvalue problems on domains with nontrivial genus and boundary. We highlight recent progress on computational approaches for extremal geometric eigenvalue problems, including (i) maximizing Laplace–Beltrami eigenvalues on closed surfaces and (ii) maximizing Steklov eigenvalues on surfaces with boundary. 

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3. Extremal Eigenvalues of Weighted Steklov Problems

   https://doi.org/10.1007/s00245-026-10438-8  

   Kao, Chiu-Yen; Mohammadi, Seyyed Abbas (June 2026, Applied Mathematics & Optimization)

   Abstract We study the optimization of Steklov eigenvalues with respect to a boundary density function$$\rho $$ $\rho$ on a bounded Lipschitz domain$$\Omega \subset \mathbb {R}^N$$ $\Omega \subset R^N$ . We investigate the minimization and maximization of$$\lambda _k(\rho )$$ ${\lambda }_k\left(\rho \right)$ , thekth Steklov eigenvalue, over admissible densities satisfying pointwise bounds and a fixed integral constraint. Our analysis covers both first and higher-order eigenvalues and applies to domains$$\Omega $$ $\Omega$ with general geometry and topology. We establish the existence of optimal solutions and provide structural characterizations: minimizers are bang--bang functions and may have disconnected support, while maximizers are not necessarily bang--bang. On circular domains, the minimization problem admits infinitely many minimizers generated by rotational symmetry, while the maximization problem has infinitely many distinct maximizers that are not symmetry-induced. We also show that the maps$$\rho \mapsto \lambda _k(\rho )$$ $\rho \mapsto {\lambda }_k\left(\rho \right)$ and$$\rho \mapsto 1/\lambda _k(\rho )$$ $\rho \mapsto 1/{\lambda }_k\left(\rho \right)$ are generally neither convex nor concave, limiting the use of classical convex optimization tools. To address these challenges, we analyze the objective functional and introduce a Fréchet differentiable surrogate that enables the derivation of optimality conditions. We further design an efficient numerical algorithm, with experiments illustrating the difficulty of recovering optimal densities when they lack smoothness or exhibit oscillations. 

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4. Geometric series expansion of the Neumann–Poincaré operator: Application to composite materials

   https://doi.org/10.1017/S0956792521000127  

   CHERKAEV, ELENA; KIM, MINWOO; LIM, MIKYOUNG (May 2021, European Journal of Applied Mathematics)

   null (Ed.)

   The Neumann–Poincaré (NP) operator, a singular integral operator on the boundary of a domain, naturally appears when one solves a conductivity transmission problem via the boundary integral formulation. Recently, a series expression of the NP operator was developed in two dimensions based on geometric function theory [34]. In this paper, we investigate geometric properties of composite materials using this series expansion. In particular, we obtain explicit formulas for the polarisation tensor and the effective conductivity for an inclusion or a periodic array of inclusions of arbitrary shape with extremal conductivity, in terms of the associated exterior conformal mapping. Also, we observe by numerical computations that the spectrum of the NP operator has a monotonic behaviour with respect to the shape deformation of the inclusion. Additionally, we derive inequality relations of the coefficients of the Riemann mapping of an arbitrary Lipschitz domain using the properties of the polarisation tensor corresponding to the domain. 

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5. Computing the Tracy-Widom Distribution for Arbitrary $$\beta>0$$

   https://doi.org/10.3842/SIGMA.2024.005  

   Trogdon, Thomas; Zhang, Yiting (January 2024, Symmetry, Integrability and Geometry: Methods and Applications)

   We compute the Tracy-Widom distribution describing the asymptotic distribution of the largest eigenvalue of a large random matrix by solving a boundary-value problem posed by Bloemendal in his Ph.D. Thesis (2011). The distribution is computed in two ways. The first method is a second-order finite-difference method and the second is a highly accurate Fourier spectral method. Since $$\beta$$ is simply a parameter in the boundary-value problem, any $$\beta> 0$$ can be used, in principle. The limiting distribution of the $$n$$th largest eigenvalue can also be computed. Our methods are available in the Julia package TracyWidomBeta.jl. 

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