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

W. Farkas

Abstract: The obtaining of sharp estimates for the asymptotic behaviour of the eigenvalues of the (semi-elliptic) operator acting in the anisotropic Sobolev space $$ Ø{W}_2^{(1,2)}(\Omega) = \left\{u \in W^{(1,2)}_2(\Omega): \, u|\partial \Omega = {\partial u \over \partial x_2}| \, \partial\Omega = 0\right\} $$ generated by the quadratic form $\int_\Omega f(\gamma )\,\overline{g(\gamma)} \,d\mu(\gamma)$ is investigated. Here $\mu$ is an appropriate self-affine fractal measure on the unit disc $\Omega \subset \R^2$.

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

Classification (MSC2000): 35P15, 46E35, 28A90

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