Maximizing the Second Robin Eigenvalue of Simply Connected Curved Membranes

IF 0.6 4区 数学 Q3 MATHEMATICS Computational Methods and Function Theory Pub Date : 2023-12-26 DOI:10.1007/s40315-023-00516-1
Jeffrey J. Langford, Richard S. Laugesen
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Abstract

The second eigenvalue of the Robin Laplacian is shown to be maximal for a spherical cap among simply connected Jordan domains on the 2-sphere, for substantial intervals of positive and negative Robin parameters and areas. Geodesic disks in the hyperbolic plane similarly maximize the eigenvalue on a natural interval of negative Robin parameters. These theorems extend work of Freitas and Laugesen from the Euclidean case (zero curvature) and the authors’ hyperbolic and spherical results for Neumann eigenvalues (zero Robin parameter). Complicating the picture is the numerically observed fact that the second Robin eigenfunction on a large spherical cap is purely radial, with no angular dependence, when the Robin parameter lies in a certain negative interval depending on the cap aperture.

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最大化简单连接曲面膜的第二罗宾特征值
对于 2 球面上简单连接的约旦域中的球帽,在正负罗宾参数和面积的大区间内,罗宾拉普拉奇的第二个特征值被证明是最大的。双曲面中的测地圆盘同样在负罗宾参数的自然区间内最大化特征值。这些定理扩展了 Freitas 和 Laugesen 在欧几里得情况(零曲率)下的研究成果,以及作者在双曲面和球面情况下对诺伊曼特征值(零罗宾参数)的研究成果。使问题更加复杂的是,根据数值观察,当罗宾参数位于某个负区间时,大球盖上的第二个罗宾特征函数是纯径向的,与角度无关,这取决于球盖的孔径。
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来源期刊
Computational Methods and Function Theory
Computational Methods and Function Theory MATHEMATICS, APPLIED-MATHEMATICS
CiteScore
3.20
自引率
0.00%
发文量
44
审稿时长
>12 weeks
期刊介绍: CMFT is an international mathematics journal which publishes carefully selected original research papers in complex analysis (in a broad sense), and on applications or computational methods related to complex analysis. Survey articles of high standard and current interest can be considered for publication as well.
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