Contributions published at Quantitative Finance (Erich Walter Farkas)

Let Ω be a bounded domain with fractal boundary, for instance von Koch's snowflake domain. First we determine the range and the kernel of the trace on ∂Ω of Sobolev spaces of fractional order defined on Ω. This extends some earlier results of H. Wallin and J. Marschall Secondly we apply these results in studying Dirichlet forms related to subordinate reflecting diffusions in non–smooth domains.

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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