On oscillatory elliptic equations on manifolds
Authors:
A. Baider and E. A. Feldman
Journal:
Trans. Amer. Math. Soc. 258 (1980), 495504
MSC:
Primary 58G25; Secondary 35B05, 35J15
DOI:
https://doi.org/10.1090/S00029947198005581866
MathSciNet review:
558186
Fulltext PDF Free Access
Abstract  References  Similar Articles  Additional Information
Abstract: In this note we investigate the possibility of oscillatory behavior for a secondorder selfadjoint elliptic operators on noncompact Riemannian manifolds (M, g). Let A be such an operator which is semibounded below and symmetric on $C_0^\infty (M) \subseteq {L^2}(M, d\mu )$ where $d\mu$ is a volume element on M. If $\varphi$ is a ${C^\infty }$ function such that $A\varphi = \lambda \varphi$, we would naively say that $\varphi$ is oscillatory (and by extension $\lambda$ is oscillatory if it possesses such an eigenfunction $\varphi$) if $M  {\varphi ^{  1}}(0)$ has an infinite number of bounded connected components. For technical reasons this is not quite adequate for a definition. However, in §1 we give the usual definition of oscillatory which is a slight generalization of the one above. Let ${\Lambda _0}$ be the number below which this phenomenon cannot occur; ${\Lambda _0}$ is the oscillatory constant for the operator A. In that A is semibounded and symmetric on $C_0^\infty (M) \subseteq {L^2}(M, d\mu )$, A has a Friedrichs extension. Let ${\Lambda _c}$ be the bottom of the continuous spectrum of the Friedrichs extension of A. Our main result is ${\Lambda _0} = {\Lambda _c}$.

A. Baider, Noncompact Riemannian manifolds with discrete spectra, J. Differential Geometry (to appear).
 Marcel Berger, Paul Gauduchon, and Edmond Mazet, Le spectre d’une variété riemannienne, Lecture Notes in Mathematics, Vol. 194, SpringerVerlag, BerlinNew York, 1971 (French). MR 0282313
 I. Chavel and E. A. Feldman, Spectra of domains in compact manifolds, J. Functional Analysis 30 (1978), no. 2, 198–222. MR 515225, DOI https://doi.org/10.1016/00221236%2878%29900708
 Shiu Yuen Cheng, Eigenfunctions and nodal sets, Comment. Math. Helv. 51 (1976), no. 1, 43–55. MR 397805, DOI https://doi.org/10.1007/BF02568142
 R. Courant and D. Hilbert, Methods of mathematical physics. Vol. I, Interscience Publishers, Inc., New York, N.Y., 1953. MR 0065391
 Philip Hartman and Calvin R. Putnam, The least cluster point of the spectrum of boundary value problems, Amer. J. Math. 70 (1948), 849–855. MR 27928, DOI https://doi.org/10.2307/2372216
 John Piepenbrink, Nonoscillatory elliptic equations, J. Differential Equations 15 (1974), 541–550. MR 342829, DOI https://doi.org/10.1016/00220396%2874%29900722 J. Rauch, Partial differential equations and related topics, Lecture Notes in Math., vol. 446, SpringerVerlag, Berlin and New York, 1975, pp. 354389.
 Jeffrey Rauch and Michael Taylor, Potential and scattering theory on wildly perturbed domains, J. Functional Analysis 18 (1975), 27–59. MR 377303, DOI https://doi.org/10.1016/00221236%2875%29900282 F. Riesz and B. Nagy, Functional analysis, Ungar, New York, 1955.
 David A. Stone, Geodesics in piecewise linear manifolds, Trans. Amer. Math. Soc. 215 (1976), 1–44. MR 402648, DOI https://doi.org/10.1090/S00029947197604026488
 Hassler Whitney, Geometric integration theory, Princeton University Press, Princeton, N. J., 1957. MR 0087148
Retrieve articles in Transactions of the American Mathematical Society with MSC: 58G25, 35B05, 35J15
Retrieve articles in all journals with MSC: 58G25, 35B05, 35J15
Additional Information
Article copyright:
© Copyright 1980
American Mathematical Society