The Robin Laplacian—Spectral conjectures, rectangular theorems
The Robin Laplacian—Spectral conjectures, rectangular theorems
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DOI：
10.1063/1.5116253
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发表时间：
2019-05
影响因子：
1.3
通讯作者：
R. Laugesen
中科院分区：
文献类型：
--
作者：
R. Laugesen
形状优化的前两个特征值的罗宾拉普拉斯算子的开发和支持新的结果为矩形盒。当Robin参数α∈R按周长缩放时，平方最小化面积归一化下矩形间的第一特征值;对于α值的尖锐范围，平方最大化第二特征值;对于每个α∈R，线段最小化直径归一化下的Robin谱间隙;当α > 0时，平方最大化矩形间的谱比。此外，每个固定矩形的谱隙是α的增函数;第二个特征值是凹的，并且除了Neumann情况外，矩形的形状只能从它的前两个频率听到. Robin Laplacian的前两个特征值的形状优化图得到了发展，并得到了矩形盒的新结果的支持.当Robin参数α∈R按周长缩放时，平方最小化面积归一化下矩形间的第一特征值;对于α值的尖锐范围，平方最大化第二特征值;对于每个α∈R，线段最小化直径归一化下的Robin谱间隙;当α > 0时，平方最大化矩形间的谱比。此外，每个固定矩形的频谱间隙是α的增函数;第二个特征值是凹的，并且，除了诺依曼的情况，矩形的形状可以从它的前两个频率听到。
Shape optimization conjectures for the first two eigenvalues of the Robin Laplacian are developed and supported with new results for rectangular boxes. The square minimizes the first eigenvalue among rectangles under area normalization when the Robin parameter α∈R is scaled by perimeter; the square maximizes the second eigenvalue for a sharp range of α-values; the line segment minimizes the Robin spectral gap under diameter normalization for each α∈R; and the square maximizes the spectral ratio among rectangles when α > 0. Furthermore, the spectral gap of each fixed rectangle is an increasing function of α; the second eigenvalue is concave, and, except in the Neumann case, the shape of the rectangle can be heard from just its first two frequencies.Shape optimization conjectures for the first two eigenvalues of the Robin Laplacian are developed and supported with new results for rectangular boxes. The square minimizes the first eigenvalue among rectangles under area normalization when the Robin parameter α∈R is scaled by perimeter; the square maximizes the second eigenvalue for a sharp range of α-values; the line segment minimizes the Robin spectral gap under diameter normalization for each α∈R; and the square maximizes the spectral ratio among rectangles when α > 0. Furthermore, the spectral gap of each fixed rectangle is an increasing function of α; the second eigenvalue is concave, and, except in the Neumann case, the shape of the rectangle can be heard from just its first two frequencies.