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Some results related to Schiffer’s problem

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Abstract

We consider the following semilinear overdetermined problem on a two dimensional bounded or unbounded domain Ω with analytic boundary ∂Ω having at least one bounded connected component

$$\left\{ {\begin{array}{*{20}{c}} { - \Delta u = g(u)}\;\;\;\;\;\;\;&{\text{in}\;\Omega ,} \\ {\tfrac{{\partial u}}{{\partial v}} = 0\;\text{and}\;u = c}&{\text{on}\;\partial \Omega ,} \end{array}} \right.$$

where c is a constant. When g(c) = 0 the constant solution u ≡ c is the unique solution. For g(c) ≠ 0, we show that the boundary is a circle if and only if the problem admits a solution that has a constant third or fourth normal derivative along the boundary. A similar result involving the fifth normal derivative is proved.

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Acknowledgements

This research was begun during a “Research in Pairs” stay from May 23 to June 14, 2013 at Mathematisches Forschungsinstitut Oberwolfach. We are grateful to MFO and their staff for the excellent working conditions and hospitality. It was then continued in 2016 and 2017 with a grant from Cologne University and a Humboldt fellowship for the second author. The second author has also been supported by MINECO grants MTM2014 and MTM2017. Finally, we would like to thank W. Reichel for a helpful discussion on the exterior problem in July 2018.

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Authors and Affiliations

  1. Mathematisches Institut, Universitat zu Koln, D-50923, Koln, Germany

    Bernd Kawohl

  2. Mathematics Department, The City University of New York CSI, Staten Island, NY, 10314, USA

    Marcello Lucia

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  1. Bernd Kawohl
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  2. Marcello Lucia
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Correspondence to Bernd Kawohl.

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Kawohl, B., Lucia, M. Some results related to Schiffer’s problem. JAMA 142, 667–696 (2020). https://doi.org/10.1007/s11854-020-0146-z

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  • DOI: https://doi.org/10.1007/s11854-020-0146-z

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