# Eigenvalue distribution of some fractal semi-elliptic differential operators

Farkas, Walter (2001). Eigenvalue distribution of some fractal semi-elliptic differential operators. Mathematische Zeitschrift, 236(2):291-320.

## 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}} , x&=& (x_1,x_2)\in \R ^2 ,

and Sierpinski carpets \G⊂\R$^2$. The aim of the paper is to investigate spectral properties of the fractal differential operator A$^{−1}$∘tr$^Γ$ acting in the anisotropic Sobolev space W$^{(t1,t2)}$2(R$^2$) where tr$^Γ$ is closely related to the trace operator tr$_Γ$.

## 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}} , x&=& (x_1,x_2)\in \R ^2 ,

and Sierpinski carpets \G⊂\R$^2$. The aim of the paper is to investigate spectral properties of the fractal differential operator A$^{−1}$∘tr$^Γ$ acting in the anisotropic Sobolev space W$^{(t1,t2)}$2(R$^2$) where tr$^Γ$ is closely related to the trace operator tr$_Γ$.

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## Additional indexing

Item Type: Journal Article, refereed, original work 03 Faculty of Economics > Department of Banking and Finance 330 Economics Physical Sciences > General Mathematics English 2001 24 Oct 2014 11:48 30 Nov 2022 14:21 Springer 0025-5874 Closed Publisher DOI. An embargo period may apply. https://doi.org/10.1007/PL00004832 http://www.math.ethz.ch/~farkas/research/F.pdf merlin-id:4262