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.
Similar content being viewed by others
References
C. Bandle,Isoperimetric Inequalities and Applications. Pitman, London 1980.
A. Bennett,Symmetry in an overdetermined fourth order elliptic boundary value problem. SIAM J. Math. Anal.17, 1354–1358 (1986).
J. R. Kuttler and V. G. Sigillito,Inequalities for membrane and Stekloff eigenvalues. J. Math. Anal. and Appl.23, 148–160 (1968).
L. E. Payne and P. W. Schaefer,Duality theorems in some overdetermined boundary value problems. Math. Meth. in the Appl. Sci11, 805–819 (1989).
G. Pólya and G. Szegö,Isoperimetric Inequalities in Mathematical Physics. Princeton Univ. Press, Princeton 1951.
M. Schiffer and G. Szegö,Virtual Mass and Polarization. Trans. Amer. Math. Soc.67, 130–205 (1949).
J. B. Serrin,A symmetry problem in potential theory. Arch. Rat. Mech. Anal.43, 304–318 (1971).
M. W. Stekloff,Sur les problèmes fondamentaux en physique mathématique. Ann. Sci. École Norm. Sup.19, 455–490 (1902).
H. F. Weinberger,Remark on the preceding paper of Serrin. Arch. Rat. Mech. Anal.43, 319–320 (1971).
Author information
Authors and Affiliations
Rights and permissions
About this article
Cite this article
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
Received:
Issue Date:
DOI: https://doi.org/10.1007/BF00944568