zbMATH — the first resource for mathematics

The double layer potentials for a bounded domain with fractal boundary. (English) Zbl 0854.31001
Král, Josef (ed.) et al., Potential theory – ICPT ’94. Proceedings of the international conference, Kouty, Czech Republic, August 13–20, 1994. Berlin: deGruyter. 463-471 (1996).
For a bounded domain $$D$$ in $$\mathbb{R}^d$$ with fractal boundary (of Hausdorff dimension greater than $$d-1$$) the author defines the double layer Newton potential $$\Phi f$$ of an $$\alpha$$-Hölder continuous function $$f$$ on $$\partial D$$ $$(f\in \Lambda_\alpha (\partial D))$$. The main result concerns the boundary behaviour of the double layer potentials. For example, under certain conditions $$\Phi f$$ is harmonic on $$\mathbb{R}^d \setminus \partial D$$ and for every $$z\in \partial D$$ $\lim_{x\to z,\;x\in D} \Phi f(x)= \int_{\mathbb{R}^d \setminus \overline {D}} \langle \nabla_y {\mathcal E} (f) (y), \nabla_y N(z- y)\rangle dy,$ where $${\mathcal E}$$ is a certain linear operator that maps $$\Lambda_\alpha (\partial D)$$ into $$\Lambda_\alpha (\mathbb{R}^d)$$, and $$N$$ is the Newton kernel. A similar result is valid for the case when $$x$$ converges to $$z$$ from the exterior of $$W$$.
For the entire collection see [Zbl 0844.00023].

MSC:
 31B25 Boundary behavior of harmonic functions in higher dimensions 28A80 Fractals