Maximizing the Second Robin Eigenvalue of Simply Connected Curved Membranes
Maximizing the Second Robin Eigenvalue of Simply Connected Curved Membranes
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DOI：
10.1007/s40315-023-00516-1
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发表时间：
2023-05
影响因子：
2.1
通讯作者：
J. Langford;R. Laugesen
中科院分区：
文献类型：
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作者：
J. Langford;R. Laugesen
罗宾拉普拉斯算子的第二个特征值对于 2-球上简单连接的 Jordan 域中的球盖来说是最大的，对于正负 Robin 参数和面积的显着区间。双曲平面中的测地盘同样会在负 Robin 参数的自然区间上最大化特征值。这些定理扩展了 Freitas 和 Laugesen 从欧几里得情况（零曲率）以及作者的诺伊曼特征值（零 Robin 参数）的双曲和球面结果的工作。使情况变得复杂的是数值观察到的事实，即当 Robin 参数位于取决于球盖孔径的某个负区间时，大球盖上的第二 Robin 特征函数是纯径向的，与角度无关。
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.