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In this paper a rearrangement minimization problem corresponding to solutions of the p -Laplacian equation is considered. The solution of the minimization problem determines the optimal way of exerting external forces on a membrane in order to have a minimum displacement. Geometrical and topological properties of the optimizer is derived and the analytical solution of the problem is obtained for circular and annular membranes. Then, we find nearly optimal solutions which are shown to be good approximations to the minimizer for specific ranges of the parameter values in the optimization problem. A robust and efficient numerical algorithm is developed based upon rearrangement techniques to derive the solution of the minimization problem for domains with different geometries in ℝ 2 and ℝ 3 . 

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. 

Purpose. This study investigated how a conscious change in ocular accommodation affects intraocular pressure (IOP) and ocular biometrics in healthy adult volunteers of different ages. Methods. Thirty-five healthy volunteers without ocular disease or past ocular surgery, and with refractive error between −3.50 and +2.50 diopters, were stratified into 20, 40, and 60 year old (y.o.) age groups. Baseline measurements of central cornea thickness, anterior chamber depth, anterior chamber angle, cornea diameter, pupil size, and ciliary muscle thickness were made by autorefraction and optical coherence tomography (OCT), while IOP was measured by pneumotonometry. Each subject’s right eye focused on a target 40 cm away. Three different tests were performed in random order: (1) 10 minutes of nonaccommodation (gazing at the target through lenses that allowed clear vision without accommodating), (2) 10 minutes of accommodation (addition of a minus 3 diopter lens), and (3) 10 minutes of alternating between accommodation and nonaccommodation (1-minute intervals). IOP was measured immediately after each test. A 20-minute rest period was provided between tests. Data from 31 subjects were included in the study. ANOVA and paired t-tests were used for statistical analyses. Results. Following alternating accommodation, IOP decreased by 0.7 mmHg in the right eye when all age groups were combined ( p  = 0.029). Accommodation or nonaccommodation alone did not decrease IOP. Compared to the 20 y.o. group, the 60 y.o. group had a thicker ciliary muscle within 75 μm of the scleral spur, a thinner ciliary muscle at 125–300 μm from the scleral spur, narrower anterior chamber angles, shallower anterior chambers, and smaller pupils during accommodation and nonaccommodation ( p ’s < 0.01). Conclusion. Alternating accommodation, but not constant accommodation, significantly decreased IOP. This effect was not lost with aging despite physical changes to the aging eye. A greater accommodative workload and/or longer test period may improve the effect. 

In this paper, spectral methods based on conformal mappings are proposed to solve the Steklov eigenvalue problem and its related shape optimization problems in two dimensions. To apply spectral methods, we first reformulate the Steklov eigenvalue problem in the complex domain via conformal mappings. The eigenfunctions are expanded in Fourier series so the discretization leads to an eigenvalue problem for coefficients of Fourier series. For shape optimization problem, we use a gradient ascent approach to find the optimal domain which maximizes k-th Steklov eigenvalue with a fixed area for a given k. The coefficients of Fourier series of mapping functions from a unit circle to optimal domains are obtained for several different k.