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Analyticity of Steklov eigenvalues of nearly circular and nearly spherical domains

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Abstract

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.

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Notes

  1. 1.

    Throughout this paper, we use the notation \(\mathbb {N}\) to denote the positive integers.

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Acknowledgements

We would like to thank Nilima Nigam, Fadil Santosa, and Chee Han Tan for helpful discussions. B. Osting is partially supported by NSF DMS 16-19755 and 17-52202. We would also like to thank the referees for their helpful comments.

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Correspondence to Braxton Osting.

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Viator, R., Osting, B. Analyticity of Steklov eigenvalues of nearly circular and nearly spherical domains. Res Math Sci 7, 4 (2020). https://doi.org/10.1007/s40687-020-0202-4

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Keywords

  • Dirichlet-to-Neumann operator
  • Steklov eigenvalues
  • Perturbation theory

Mathematics Subject Classification

  • 26E05
  • 35C20
  • 35P05
  • 41A58