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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
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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.


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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