Mathematical Proceedings of the Cambridge Philosophical Society

Research Article

Counterexamples to the modified Weyl–Berry conjecture on fractal drums

Michel L. Lapidusa1* and Carl Pomerancea2*

a1 Department of Mathematics, Sproul Hall, University of California, Riverside, CA 92521-0135, U.S.A, e-mail: lapidus@math.ucr.edu

a2 Department of Mathematics, University of Georgia, Boyd Graduate Studies Research Center, Athens, GA 30602, U.S.A, e-mail: carl@ada.math.uga.edu

Let Ω be a non-empty open set in xs211Dn with finite ‘volume’ (n-dimensional Lebesgue measure). Let S0305004100074053_inline1 be the Laplacian operator. Consider the eigenvalue problem (with Dirichlet boundary conditions):

S0305004100074053_eqn001

where λ xs2208 xs211D and u is a non-zero member of S0305004100074053_inline2 (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 λxs2208xs211D for which (1·1) has a non-zero solution S0305004100074053_inline3 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 S0305004100074053_inline4 where 0 < λ1 ≤ λ2 ≤ … and where the multiplicity of each λ in the sequence is the dimension of the associated eigenspace. Let

S0305004100074053_eqnU001

(Received December 14 1993)

(Revised September 26 1994)

Footnotes

* Supported in part by NSF grants DMS-9207098 (M.L.L.) and DMS-9206784 (C.P.).