Neumann特征值的Wolf-Keller定理
Wolf-Keller theorem for Neumann eigenvalues
摘要：
经典的Schney-Weinberger不等式指出，在给定面积的有界平面区域中，第一个非零Neumann特征值被圆盘最大化。最近，Girouard，Nadirashvili和Polterovich证明了，对于给定面积的单连通平面区域，第二个非零Neumann特征值在极限下通过退化为两个相同圆盘的不相交并的区域序列而最大化。我们证明了固定面积的平面区域的Neumann特征值不总是由任意圆盘的不相交并集最大化。这是一个类似的结果沃尔夫和凯勒证明较早的狄利克雷特征值。
英文摘要：
The classical Szego-Weinberger inequality states that among bounded planar domains of given area, the first nonzero Neumann eigenvalue is maximized by a disk. Recently, it was shown by Girouard, Nadirashvili and Polterovich that, for simply connected planar domains of given area, the second nonzero Neumann eigenvalue is maximized in the limit by a sequence of domains degenerating to a disjoint union of two identical disks. We prove that Neumann eigenvalues of planar domains of fixed area are not always maximized by a disjoint union of arbitrary disks. This is an analogue of a result by Wolf and Keller proved earlier for Dirichlet eigenvalues.
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