Domain-independent upper bounds for eigenvalues of elliptic operators

Author:
Stephen M. Hook

Journal:
Trans. Amer. Math. Soc. **318** (1990), 615-642

MSC:
Primary 35J25; Secondary 35P15, 47F05

MathSciNet review:
994167

Full-text PDF Free Access

Abstract | References | Similar Articles | Additional Information

Abstract: Let be a bounded open set, its boundary and the Laplacian on . Consider the elliptic differential equation:

(1)

(2)

In this paper we abstract the method used by Hile and Protter [2] to establish (2) and apply the method to a variety of second-order elliptic problems, in particular, to all constant coefficient problems. We then consider a variety of higher-order problems and establish an extension of (2) for problem (1) where the Laplacian is replaced by a more general operator in a Hilbert space.

**[1]**Zu Chi Chen,*Inequalities for eigenvalues of a class of polyharmonic operators*, Appl. Anal.**27**(1988), no. 4, 289–314. MR**936473**, 10.1080/00036818808839742**[2]**G. N. Hile and M. H. Protter,*Inequalities for eigenvalues of the Laplacian*, Indiana Univ. Math. J.**29**(1980), no. 4, 523–538. MR**578204**, 10.1512/iumj.1980.29.29040**[3]**G. N. Hile and R. Z. Yeh,*Inequalities for eigenvalues of the biharmonic operator*, Pacific J. Math.**112**(1984), no. 1, 115–133. MR**739143****[4]**S. M. Hook,*Inequalities for eigenvalues of self-adjoint operators*, Doctoral Dissertation, University of California, Berkeley, 1986.**[5]**Stephen M. Hook,*Inequalities for eigenvalues of selfadjoint operators*, Trans. Amer. Math. Soc.**318**(1990), no. 1, 237–259. MR**943604**, 10.1090/S0002-9947-1990-0943604-8**[6]**-,*Bounds for the fundamental frequencies of an elastic medium*, Preprint.**[7]**Bernhard Kawohl and Guido Sweers,*Remarks on eigenvalues and eigenfunctions of a special elliptic system*, Z. Angew. Math. Phys.**38**(1987), no. 5, 730–740 (English, with German summary). MR**917475**, 10.1007/BF00948293**[8]**L. E. Payne, G. Pólya, and H. F. Weinberger,*On the ratio of consecutive eigenvalues*, J. Math. and Phys.**35**(1956), 289–298. MR**0084696**

Retrieve articles in *Transactions of the American Mathematical Society*
with MSC:
35J25,
35P15,
47F05

Retrieve articles in all journals with MSC: 35J25, 35P15, 47F05

Additional Information

DOI:
http://dx.doi.org/10.1090/S0002-9947-1990-0994167-2

Article copyright:
© Copyright 1990
American Mathematical Society