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Eigenvalue asymptotics for an elliptic boundary problem

Published online by Cambridge University Press:  26 March 2007

M. Faierman  and
M. Möller
M. Faierman
Affiliation:
School of Mathematics, The University of New South Wales, Sydney, NSW 2052, Australia
M. Möller
Affiliation:
The John Knopfmacher Centre for Applicable Analysis and Number Theory, School of Mathematics, University of the Witwatersrand, Johannesburg, WITS 2050, South Africa ([email protected])
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Abstract

We consider an elliptic boundary problem in a bounded region Ω ⊂ ℝn wherein the spectral parameter is multiplied by a real-valued weight function with the property that it, together with its reciprocal, is essentially bounded in Ω. The problem is considered under limited smoothness assumptions and under an ellipticity with parameter condition. Then, fixing our attention upon the operator induced on L2(Ω) by the boundary problem under null boundary conditions, we establish results pertaining to the asymptotic behaviour of the eigenvalues of this operator under weaker smoothness assumptions than have hitherto been supposed.

Type
Research Article
Information
Proceedings of the Royal Society of Edinburgh Section A: Mathematics , Volume 137 , Issue 2 , 2007 , pp. 281 - 302
Copyright
Copyright © Royal Society of Edinburgh 2007

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References

1 Adams, R. A.. Sobolev spaces (Academic, 1975).Google Scholar
2 Agmon, S.. On the eigenfunctions and on the eigenvalues of general elliptic boundary value problems. Commun. Pure Appl. Math. 15 (1962), 119–147.Google Scholar
3 Agmon, S.. On kernels, eigenvalues, and eigenfunctions of operators related to elliptic problems. Commun. Pure Appl. Math. 18 (1965), 627–663.Google Scholar
4 Agmon, S.. Lectures on elliptic boundary value problems (Princeton, NJ: Van Nostrand, 1965).Google Scholar
5 Agranovich, M. S. and Markus, A. S.. On spectral properties of elliptic pseudodifferential operators far from selfadjoint ones. Z. Analysis Anwend. 8 (1989), 237–260.Google Scholar
6 Agranovich, M. S., Denk, R. and Faierman, M.. Weakly smooth nonselfadjoint elliptic boundary problems. In Spectral theory, microlocal analysis, singular manifolds: advances in partial differential equations, Mathematical Topics, vol. 14, pp. 138–199 (Berlin: Akademie. 1997).Google Scholar
7 Beals, R.. Asymptotic behaviour of the Green's function of an elliptic operator. J. Fund. Analysis 5 (1970), 484–503.CrossRefGoogle Scholar
8 Denk, R., Faierman, M. and Möller, M.. An elliptic boundary problem for a system involving a discontinuous weight. Manuscr. Math. 108 (2001), 289–317.Google Scholar
9 Duistermaat, J. J.. On operators of trace class in L2 (X, μ). Proc. Indian Acad. Sci. Math. Sci. 90 (1981), 29–32.Google Scholar
10 Dunford, N. and Schwartz, J. T.. Linear operators, part I (Wiley, 1988).Google Scholar
11 Faicrman, M.. Eigenvalue asymptotics for an elliptic boundary problem involving an indefinite weight. Integ. Eqns Operat. Theory 43 (2002), 131–154.Google Scholar
12 Grisvard, P.. Elliptic problems in nonsmooth domains (London: Pitman, 1985).Google Scholar
13 Maz'ja, V. G.. Sobolev spaces (Springer, 1985).Google Scholar
14 Mizohata, S.. Sur les propriétés asymptotiques des valours propres pour les opérateurs elliptiques. J. Math. Kyoto Univ. 4 (1965), 399–428.Google Scholar
15 Triebel, H.. Interpolation theory, function spaces, differential operators (Amsterdam: North-Holland, 1978).Google Scholar