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Zeitschrift für Analysis und ihre Anwendungen


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Volume 18, Issue 4, 1999, pp. 875–893
DOI: 10.4171/ZAA/920

Published online: 1999-12-31

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

W. Farkas[1]

(1) Universität der Bundeswehr München, Neubiberg, Germany

The obtaining of sharp estimates for the asymptotic behaviour of the eigenvalues of the (semi-elliptic) operator acting in the anisotropic Sobolev space $$W_2^{(1,2)}(\Omega) = \{ u \in W_2^{(1,2)} (\Omega) : u|\partial \Omega = \frac{\partial u}{\partial x_2}| \partial \Omega = 0 \}$$ generated by the quadratic form $\int_{\Omega} f(\gamma) g (\gamma)d \mu (\gamma)$ is investigated. Here $\mu$ is an appropriate self-affine fractal measure on the unit disc $\Omega \subset \mathbb R^2$.

Keywords: Regular anisotropic fractals, anisotropic function spaces, semi-elliptic differential operators

Farkas W.: The Behaviour of the Eigenvalues for a Class of Operators Related to some Self-Affine Fractals in $\mathbb R^2$. Z. Anal. Anwend. 18 (1999), 875-893. doi: 10.4171/ZAA/920