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Some overdetermined boundary value problems for harmonic functions

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Summary

In this paper we study various overdetermined problems involving harmonic functions. In particular, we show that if the second eigenfunctionu 2 of the Stekloff eigenvalue problem in a bounded simply connected plane domain Ω has a constant value of ∥∂u 2∥ on ∇Ω, then Ω is a disk

Résumé

Cet article est consacré à l'étude de certains problèmes surdéterminés pour des fonctions harmoniques. En particulier, nous montrons que si le gradient de la seconde fonction propre du problème de Stekloff défini dans un domaine Ω borné, simplement connexe du plan, a son module constant sur la frontière ∂Ω, alors Ω est nécessairement un disque.

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Authors and Affiliations

  1. Dept of Mathematics, Cornell University, 14853, Ithaca, N.Y., USA

    L. E. Payne

  2. Université Laval, G1K 7P4, Québec, Canada

    G. A. Philippin

Authors
  1. L. E. Payne
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  2. G. A. Philippin
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Payne, L.E., Philippin, G.A. Some overdetermined boundary value problems for harmonic functions. Z. angew. Math. Phys. 42, 864–873 (1991). https://doi.org/10.1007/BF00944568

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  • DOI: https://doi.org/10.1007/BF00944568

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