We consider the Dirichlet-to-Neumann operator (DNO) on nearly circular and nearly spherical domains in two and three dimensions, respectively. Treating such domains as perturbations of the ball, we prove the analyticity of the DNO with respect to the domain perturbation parameter. Consequently, the Steklov eigenvalues are also shown to be analytic in the domain perturbation parameter. To obtain these results, we use the strategy of Nicholls and Nigam (J Comput Phys 194(1):278–303, 2004. https://doi.org/10.1016/j.jcp.2003.09.006); we transform the equation on the perturbed domain to a ball and geometrically bound the Neumann expansion of the transformed Dirichlet-to-Neumann operator.

中文翻译：

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近似圆形和近似球形域的Steklov特征值的解析性

我们分别在二维和三维的近圆形和近球形域上考虑Dirichlet-to-Neumann算子（DNO）。将此类域视为球的摄动，我们证明了DNO相对于域摄动参数的解析性。因此，Steklov特征值在域摄动参数中也被证明是解析的。为了获得这些结果，我们使用了Nicholls和Nigam的策略（J Comput Phys 194（1）：278-303，2004. https://doi.org/10.1016/j.jcp.2003.09.006）；我们将扰动域上的方程转换为一个球，并将转换后的Dirichlet-to-Neumann算子的Neumann展开几何约束。