A constructive approximation theorem for the oblique derivative problem in potential theory
Willi Freeden, Heinrich Kersten, Rolf Leis · Mathematical Methods in the Applied Sciences · 1981
Abstract The (exterior) oblique derivative problem of potential theory magnified image is considered where l is a C(1,λ)‐vector field on a regular boundary ∂Ge in Euclidean space R3 and the direction l at any point of ∂Ge forms with the outside normal n an angle ∢ (l, n) satisfying cos ∢ (l, n) ≥ c > 0. An approximation of the uniquely determined solution is given by use of trial functions {Φn}n=0,1,… harmonic in some region containing Ge and suitable for numerical purpose (for instance: solid spherical harmonics, certain sequences of fundamental solutions). The system {l▽Φn+hΦn}n=0,1,… defined on ∂Ge is shown to be closed and complete in the pre‐Hilbert space C(0,λ)(∂Ge) of λ‐Hölder continuous functions.