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].


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