We consider the Dirichlet-to-Neumann operator (DNO) on nearly hyperspherical domains in dimension >3\documentclass[12pt]{minimal}
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\begin{document}$$> 3$$\end{document}. Treating such domains as perturbations of the ball, we prove the analytic dependence of the DNO on the shape perturbation parameter for fixed perturbation functions. Consequently, we conclude that the Steklov eigenvalues are analytic in the shape perturbation parameter as well. 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), and Viator and Osting (Proc R Soc A 474(2220):20180072, 2018. https://doi.org/10.1098/rspa.2018.0072); we transform the Laplace-Dirichlet problem on the perturbed domain to a more complicated, parameter-dependent equation on the ball, and then geometrically bound the Neumann expansion of the transformed DNO. These results are a generalization of the work of Viator and Osting (2020) for dimension 2 and 3.
