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

摘要: Let Γ be the boundary of a simply connected bounded polyhedron Ω, in ℝ3. The harmonic double layer potential operator on is defined by $$ Ku(x): = \frac{1} {{2\pi }}\int_\Gamma {u\left( y \right)\frac{\partial } {{\partial n_y }}\frac{1} {{|x - y|}}do\left( \right)} {\frac{{(x y).\,n_y }} y|^3 }}u\left( \right)do\left( \right),\,x\, \in \,\Gamma $$ (1) where do surface measure and n outward pointing normal vector to Γ.