JournalszaaVol. 18, No. 4pp. 875–893

The Behaviour of the Eigenvalues for a Class of Operators Related to some Self-Affine Fractals in R2\mathbb R^2

  • W. Farkas

    Universität der Bundeswehr München, Neubiberg, Germany
The Behaviour of the Eigenvalues for a Class of Operators Related to some Self-Affine Fractals in $\mathbb R^2$ cover
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Abstract

The obtaining of sharp estimates for the asymptotic behaviour of the eigenvalues of the (semi-elliptic) operator acting in the anisotropic Sobolev space

W2(1,2)(Ω)={uW2(1,2)(Ω):uΩ=ux2Ω=0}W_2^{(1,2)}(\Omega) = \{ u \in W_2^{(1,2)} (\Omega) : u|\partial \Omega = \frac{\partial u}{\partial x_2}| \partial \Omega = 0 \}

generated by the quadratic form Ωf(γ)g(γ)dμ(γ)\int_{\Omega} f(\gamma) g (\gamma)d \mu (\gamma) is investigated. Here μ\mu is an appropriate self-affine fractal measure on the unit disc ΩR2\Omega \subset \mathbb R^2.

Cite this article

W. Farkas, The Behaviour of the Eigenvalues for a Class of Operators Related to some Self-Affine Fractals in R2\mathbb R^2. Z. Anal. Anwend. 18 (1999), no. 4, pp. 875–893

DOI 10.4171/ZAA/920