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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
Communities & Collections:03 Faculty of Economics > Department of Banking and Finance
Dewey Decimal Classification:330 Economics
Scopus Subject Areas:Physical Sciences > General Mathematics
Language:English
Date:2001
Deposited On:24 Oct 2014 11:48
Last Modified:30 Nov 2022 14:21
Publisher:Springer
ISSN:0025-5874
OA Status:Closed
Free access at:Publisher DOI. An embargo period may apply.
Publisher DOI:https://doi.org/10.1007/PL00004832
Official URL:http://www.math.ethz.ch/~farkas/research/F.pdf
Other Identification Number:merlin-id:4262