Self-similar drums and generalized Weierstrass functions

Jürgen Gerling and Heinz-Jürgen Schmidt

Physica A: Statistical Mechanics and its Applications, 1992, vol. 191, issue 1, 536-539

Abstract: We consider the number N(λ;Ω) of eigenvalues less than λ of the negative Laplacian with Dirichlet boundary conditions in a domain Ω⊂Rn with fractal boundary ∂Ω. It is known that for λ→∞, N(λ;Ω)=Cn|Ω|nλn/2+O(λD/2), where D is the Minkowski dimension of Ω. For a certain class of self-similar domains (“drums”) we obtain for N(λ;Ω) a second term of the form - F(ln λ)λD/2 with a bounded periodic function F.F contains a generalized Weierstrass function with a self-similar fractal graph. A number of examples with n=1,2,3,… has been studied, where more information about F is available. Finally, a possible physical application will be sketched.

Date: 1992
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Persistent link: https://EconPapers.repec.org/RePEc:eee:phsmap:v:191:y:1992:i:1:p:536-539

DOI: 10.1016/0378-4371(92)90578-E

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