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Counterexamples to the modified Weyl–Berry conjecture on fractal drums

Published online by Cambridge University Press:  24 October 2008

Michel L. Lapidus  and
Carl Pomerance
Michel L. Lapidus
Affiliation:
Department of Mathematics, Sproul Hall, University of California, Riverside, CA 92521-0135, U.S.A, e-mail: [email protected]
Carl Pomerance
Affiliation:
Department of Mathematics, University of Georgia, Boyd Graduate Studies Research Center, Athens, GA 30602, U.S.A, e-mail: [email protected]
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Extract

Let Ω be a non-empty open set in ℝn with finite ‘volume’ (n-dimensional Lebesgue measure). Let be the Laplacian operator. Consider the eigenvalue problem (with Dirichlet boundary conditions):

where λ ∈ ℝ and u is a non-zero member of (the closure in the Sobolev space H1(Ω) of the set of smooth functions with compact support contained in Ω). It is well known that the values of λ∈ℝ for which (1·1) has a non-zero solution are positive and form a discrete set. Moreover, for each λ, the associated eigenspace is finite dimensional. Let the spectrum of (1·1) be denoted where 0 < λ1 ≤ λ2 ≤ … and where the multiplicity of each λ in the sequence is the dimension of the associated eigenspace. Let

Type
Research Article
Information
Mathematical Proceedings of the Cambridge Philosophical Society , Volume 119 , Issue 1 , January 1996 , pp. 167 - 178
Copyright
Copyright © Cambridge Philosophical Society 1996

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References

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