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Asymptotic Variational Formulae for Eigenvalues

Published online by Cambridge University Press:  20 November 2018

C.A. Swanson
C.A. Swanson*
Affiliation:
The University of British Columbia
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The eigenvalues of a second order self-adjoint elliptic differential operator on Riemannian n-space R will be considered. Our purpose is to obtain asymptotic variational formulae for the eigenvalues under the topological deformations of (i) removing an ɛ -cell (and adjoining an additional boundary condition on the boundary component thereby introduced); and (ii) attaching an ɛ -handle, valid on a half-open interval 0 < ɛ ≤ ɛo. In particular the formulae will exhibit the non-analytic nature of the variation. Similar variational problems for singular ordinary differential operators have been considered by the writer in [3] and [4].

Type
Research Article
Information
Canadian Mathematical Bulletin , Volume 6 , Issue 1 , January 1963 , pp. 15 - 25
Copyright
Copyright © Canadian Mathematical Society 1963

References

1. Bohnenblust, H. F., DePrima, C. R., and Swanson, C.A., Elliptic operators with perturbed domains, in preparation.Google Scholar
2. Duff, G. F. D., Partial Differential Equations, (University of Toronto, 1956).Google Scholar
3. Swanson, C. A., Differential operators with perturbed domains, J. of Math. Mech., 6 (1957) 823-846.Google Scholar
4. Swanson, C. A., On a non-analytic perturbation problem, Canad. Math. Bull. 4 (1961) 243-248.10.4153/CMB-1961-027-6Google Scholar
5. Swanson, C. A., Asymptotic estimates for limit circle problems, Pacific J. Math., 11 (1961) 1549-1559.Google Scholar
6. Swanson, C. A., On spectral estimation, Bull. Amer. Math. Soc., 68 (1962) 33-35.10.1090/S0002-9904-1962-10689-2Google Scholar
7. Schiffer, M. and Spencer, D. C., Functionals of Finite Riemann Surfaces, (Princeton University, 1954).Google Scholar