On the Double Layer Potential Operator over Polyhedral Domains: Solvability in Weighted Sobolev Spaces and Spline Approximation

Abstract

Let Γ be the boundary of a simply connected bounded polyhedron Ω, in ℝ³. The harmonic double layer potential operator on Γ is defined by

$$ Ku(x): = \frac{1} {{2\pi }}\int_\Gamma {u\left( y \right)\frac{\partial } {{\partial n_y }}\frac{1} {{|x - y|}}do\left( y \right)} = \frac{1} {{2\pi }}\int_\Gamma {\frac{{(x - y).\,n_y }} {{|x - y|^3 }}u\left( y \right)do\left( y \right),\,x\, \in \,\Gamma } $$

(1)

where do is the surface measure on Γ and n the outward pointing normal vector to Γ.