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

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 then results are known regarding the behavior of the principal eigenvalue as . These results are very sharp in case the vector has a unique asymptotically stable point in the domain where the eigenvalue problem is considered. In this paper the case where *L* is an ordinary differential operator degenerating on the boundary of is considered. Existence and uniqueness of a principal eigenvalue and eigenfunction are proved and results on the behavior of as are established.

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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