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.

中文翻译：

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Steklov 谱在哑铃域上的渐近行为

我们在哑铃域中分析 Steklov 问题的特征值和特征向量的渐近行为，哑铃域由两个由宽度消失的细管连接的 Lipschitz 集组成。所有的特征值都收缩到零，速度由宽度的某种幂驱动，该宽度乘以一维问题的特征值。在空间的二维空间中，行为与三维或更高维度有着根本的不同，极限问题具有不同的性质。这种现象是由于仅在维度 2 中，管的边界没有消失的表面量度。