Steklov eigenvalues of reflection-symmetric nearly circular planar domains
Steklov eigenvalues of reflection-symmetric nearly circular planar domains
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反射对称近圆形平面域的 Steklov 特征值
DOI：
10.1098/rspa.2018.0072
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发表时间：
2018
期刊：
影响因子：
--
通讯作者：
B. Osting
中科院分区：
文献类型：
--
作者：
Robert Viator;B. Osting
我们认为Steklov特征值的反射对称，近圆形，平面域。把这些域作为圆盘的扰动，我们得到了域扰动参数的二阶形式渐近估计。最后，我们讨论的影响等周不等式。也就是说，我们的研究结果证实了Weinstock和Brock的结果，分别指出，光盘是面积和周长约束问题的最大化。它们也支持Hersch，Payne和Schiffer的结果，即在所有等周长的开平面集合中，前两个特征值的乘积是最大的。此外，我们的研究结果意味着，光盘是不是最大化的面积限制的问题，更高的偶数Steklov特征值，建议由以前的数值结果。
We consider Steklov eigenvalues of reflection-symmetric, nearly circular, planar domains. Treating such domains as perturbations of the disc, we obtain a second-order formal asymptotic estimate in the domain perturbation parameter. We conclude with a discussion of implications for isoperimetric inequalities. Namely, our results corroborate the results of Weinstock and Brock that state, respectively, that the disc is the maximizer for the area and perimeter constrained problems. They also support the result of Hersch, Payne and Schiffer that the product of the first two eigenvalues is maximal among all open planar sets of equal perimeter. In addition, our results imply that the disc is not the maximizer of the area constrained problems for higher even numbered Steklov eigenvalues, as suggested by previous numerical results.