Eigenvalue distribution of some fractal semi-elliptic differential operators

Preprint series: 98-35, Analysis

MSC:
46E35 Sobolev spaces and other spaces of smooth'' functions, embedding theorems, trace theorems
35P15 Estimation of eigenvalues, upper and lower bounds

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