哑铃域上Steklov问题的渐近性
Asymptotic behaviour of the Steklov problem on dumbbell domains
摘要：
本文分析了由宽度为零的细管连接的两个Lipschitz集组成的哑铃形区域中Steklov问题的特征值和特征向量的渐近行为。所有的特征值都坍缩到零，速度由宽度的某个幂驱动，该幂乘以一维问题的特征值。在二维空间中，其行为与三维或更高维空间中的行为有着根本的不同，极限问题也具有不同的性质。这是因为只有在二维空间中，管的边界才没有消失面测度。
英文摘要：
We analyse the asymptotic behaviour of the eigenvalues and eigenvectors of a Steklov problem in a dumbbell domain consisting of two Lipschitz sets connected by a thin tube with vanishing width. All the eigenvalues are collapsing to zero, the speed being driven by some power of the width which multiplies the eigenvalues of a one dimensional problem. In two dimensions of the space, the behaviour is fundamentally different from the third or higher dimensions and the limit problems are of different nature. This phenomenon is due to the fact that only in dimension two the boundary of the tube has not vanishing surface measure.
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