Eigenvalue distribution of some fractal semi-elliptic differential operators

by    W. Farkas

Preprint series: 98-35, Analysis

46E35 Sobolev spaces and other spaces of ``smooth'' functions, embedding theorems, trace theorems
35P15 Estimation of eigenvalues, upper and lower bounds
28A80 Fractals, See also {58Fxx}

Abstract: We consider differential operators of type
$$ Au(x) = u(x) + (-1)^{t_1}\frac{\partial ^{2t_1} u(x)}{\partial x_1^{2t_1}}+ (-1)^{t_2}\frac{\partial ^{2t_2} u(x)}{\partial x_2^{2t_2}}, \quad x=(x_1,x_2)\in \R ^2 , $$
and Sierpinski carpets $\Gamma\subset \R ^2$.
The aim of the paper is to investigate spectral properties of the fractal differential operator $ A^{-1}\, \circ \, tr ^{\Gamma} $ acting in the anisotropic Sobolev space $W^{(t_1,t_2)}_2(\R ^2)$ where $tr ^{\Gamma}$ is closely related to the trace operator $tr_{\Gamma}$

Keywords: regular anisotropic fractal, anisotropic function space, semi - elliptic differential operator

Upload: 1999-03-01

Update: 1999-03-01

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