The generalized riemann—hilbert boundary value problem for polyharmonic functions in the sobolev spaces W 2n,P (D)

Abstract

The article discusses the construction of a polyharmonic function Ф of order n defined on a simply-connected and bounded domain D, which satisfies 2n prescribed generalized Riemann–Hilbert boundary conditions on the boundary ∂D. The 2n boundary conditions are transformed into 2n classical Riemann–Hilbert boundary value problems for holomorphic functions G j (z)Q j (z)j= 1,…,n, defined in terms of the holormorphic functions ψ p (z),ϕ p (z),p = 0,1,…,n–1, which define the n-harmonic function Ф. The latter problems are solved, using the standard methods from the literature, and their solutions are used to determine ϕ p , ψ p ) and hence Ф.