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manuscripta mathematica

, Volume 19, Issue 1, pp 47–56 | Cite as

Über das Spektrum des Laplace-Operators in einigen unbeschränkten Gebieten

  • Jukka Saranen
Article

Abstract

In this paper we study the eigenvalues of the Laplace operator Δ u on some classes of unbounded regions G. The starting-point of our study is the paper [5] of Vogelsang. Among other things he has proved that the Laplace operator with zero boundary condition has no negative eigenvalues in unbounded regions the boundary of which satisfies ν(x) · x ≦0, where ν(x) is the exterior normal. We prove in this paper similar results of the spectrum, when the condition above is suitably disturbed. Because our main interest lies in replacing the geometric condition ν(x) · x ≦0 with another condition, we have studied neither equations with higher orders nor equations with variable coefficients.

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Literatur

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    AGMON, S.: Lectures on elliptic boundary value problems. -Van Nostrand mathematical studies 2. D. Van Nostrand Company, Inc., New York-Toronto-London-Melbourne, 1965.Google Scholar
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    KÖNIG, H.: Ein einfacher Beweis des Integralsatzes von Gauß. -J.-ber. Deutsch. Math.-Verein. 66 (1964), 119–138.Google Scholar
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    PROTTER, M.H.: Unique continuation for elliptic equations. -Trans Amer. math. Soc. 95 (1960), 81–91.Google Scholar
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    RELLICH, F.: Darstellung der Eigenwerte von Δu+λu=0 durch ein Randintegral. -Math. Z. 46 (1940), 635–636.Google Scholar
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    VOGELSANG, V.: Elliptische Differentialgleichungen mit variablen Koefficienten in Gebieten mit unbeschränktem Rand. Manuscripta math. 14 (1975), 379–401.Google Scholar

Copyright information

© Springer-Verlag 1976

Authors and Affiliations

  • Jukka Saranen
    • 1
  1. 1.Mathematisches Institut SammonkatuUniversität JyväskyläJyväskylä 10Finland

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