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Linear stability estimates for Serrin’s problem via a modified implicit function theorem

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Abstract

We examine Serrin’s classical overdetermined problem under a perturbation of the Neumann boundary condition. The solution of the problem for a constant Neumann boundary condition exists provided that the underlying domain is a ball. The question arises whether for a perturbation of the constant there still are domains admitting solutions to the problem. Furthermore, one may ask whether a domain that admits a solution for the perturbed problem is unique up to translation and whether it is close to the ball. We develop a new implicit function theorem for a pair of Banach triplets that is applicable to nonlinear problems with loss of derivatives except at the point under consideration. Combined with a detailed analysis of the linearized operator, we prove the existence and local uniqueness of a domain admitting a solution to the perturbed overdetermined problem. Moreover, an optimal linear stability estimate for the shape of a domain is established.

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Acknowledgements

The authors would like to thank the anonymous referee for useful suggestions for improving the clarity of presentation.

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  1. Alexandra Gilsbach

    Present address: , Aachen, Germany

Authors and Affiliations

  1. Department of Mathematics, School of Science, Tokyo Institute of Technology, 2-12-1 Ookayama, Meguro-ku, 152-8551, Tokyo, Japan

    Alexandra Gilsbach & Michiaki Onodera

Authors
  1. Alexandra Gilsbach
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  2. Michiaki Onodera
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Correspondence to Michiaki Onodera.

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Communicated by M. del Pino.

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While working on this paper, the first author was an International Research Fellow of Japan Society for the Promotion of Science (Postdoctoral Fellowships for Research in Japan). The second author was supported in part by the Grant-in-Aid for Scientific Research (C) 20K03673, Japan Society for the Promotion of Science.

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Gilsbach, A., Onodera, M. Linear stability estimates for Serrin’s problem via a modified implicit function theorem. Calc. Var. 60, 241 (2021). https://doi.org/10.1007/s00526-021-02107-1

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  • DOI: https://doi.org/10.1007/s00526-021-02107-1

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