Remote Access Transactions of the American Mathematical Society
Green Open Access

Transactions of the American Mathematical Society

ISSN 1088-6850(online) ISSN 0002-9947(print)

 
 

 

The asymptotic behavior of the first eigenvalue of differential operators degenerating on the boundary


Authors: Allen Devinatz and Avner Friedman
Journal: Trans. Amer. Math. Soc. 234 (1977), 505-529
MSC: Primary 34B25; Secondary 35P20
DOI: https://doi.org/10.1090/S0002-9947-1977-0466720-X
MathSciNet review: 0466720
Full-text PDF

Abstract | References | Similar Articles | Additional Information

Abstract: When L is a second order ordinary or elliptic differential operator, the principal eigenvalue for the Dirichlet problem and the corresponding principal (positive) eigenfunction u are known to exist and u is unique up to normalization. If further L has the form $ \varepsilon \Sigma {a_{ij}}{\partial ^2}/\partial {x_i}\partial {x_i} + \Sigma {b_i}\partial /\partial {x_i}$ then results are known regarding the behavior of the principal eigenvalue $ \lambda = {\lambda _\varepsilon }$ as $ \varepsilon \downarrow 0$. These results are very sharp in case the vector $ ({b_i})$ has a unique asymptotically stable point in the domain $ \omega $ where the eigenvalue problem is considered. In this paper the case where L is an ordinary differential operator degenerating on the boundary of $ \omega $ is considered. Existence and uniqueness of a principal eigenvalue and eigenfunction are proved and results on the behavior of $ {\lambda _\varepsilon }$ as $ \varepsilon \downarrow 0$ are established.


References [Enhancements On Off] (What's this?)

  • [1] A. Devinatz, R. Ellis and A. Friedman, The asymptotic behavior of the first real eigenvalue of second order elliptic operators with a small parameter in the highest derivatives. II, Indiana Univ. Math. J. 23 (1973/74), 991-1011. MR 49 #9448. MR 0344709 (49:9448)
  • [2] W. H. Fleming and C. P. Tsai, Some stochastic systems depending on small parameters, Proc. Sympos. on Dynamics (to appear). MR 0651576 (58:31400)
  • [3] A. Friedman, The asymptotic behavior of the first real eigenvalue of a second order elliptic operator with a small parameter in the highest derivatives, Indiana Univ. Math. J. 22 (1972/73), 1005-1015. MR 47 #9088. MR 0320551 (47:9088)
  • [4] M. A. Krasnosel'skiĭ, Positive solutions of operator equations, Nordhoff, Groningen, 1964. MR 31 #6107. MR 0181881 (31:6107)
  • [5] D. Ludwig, Persistence of dynamical systems under random perturbations, SIAM Rev. 17 (1975), 605-640. MR 0378122 (51:14291)
  • [6] G. F. Miller, The evaluation of eigenvalues of a differential equation arising in a problem in genetics, Proc. Cambridge Philos. Soc. 58 (1962), 588-593. MR 26 #2023. MR 0144479 (26:2023)

Similar Articles

Retrieve articles in Transactions of the American Mathematical Society with MSC: 34B25, 35P20

Retrieve articles in all journals with MSC: 34B25, 35P20


Additional Information

DOI: https://doi.org/10.1090/S0002-9947-1977-0466720-X
Keywords: Eigenvalue problem, eigenvalue, eigenfunction, principal eigenvalue, principal eigenfunction, degenerate differential operator, selfadjoint operator, resolvent operator, completely continuous operator, asymptotic behavior of the first eigenvalue
Article copyright: © Copyright 1977 American Mathematical Society

American Mathematical Society