Harmonic boundary value problems in half disc and half ring

On the solvability of boundary value problems for the nonhomogeneous polyharmonic equation in a ball

“…in some plane domains. Those include Riemann, Hilbert, Dirichlet, Neumann, Schwarz and Robin problems [2][3][4][5][6][11][12][13]. All of those works are to generalize the classical integral representation theory for analytic and harmonic functions in planar domains.…”

Schwarz problem for bi-analytic functions in multiply connected domains

“…The essential tools in these investigations are the fundamental solutions of the corresponding homogeneous equations. The Green-type functions and the representations of solutions for the boundary value problems known as Dirichlet, Neumann, Robin, Schwarz and Riemann-Hilbert problems defined for such model equations are obtained in simply connected domains [1][2][3][4][5] including unbounded domains [6][7][8][9]. Representations of solutions have been extended to the boundary value problems for linear differential equations of arbitrary orders by introducing integral operators via exactly constructed Green-type functions.…”

Harmonic Dirichlet Problem in a Ring Sector

“…Many kinds of concrete applied problems led to the investigation of boundary value problems for complex partial differential equations in different domains [1][2][3][4][5][6]. In [2], the authors studied harmonic boundary value problems in half disc and half ring.…”

“…In [2], the authors studied harmonic boundary value problems in half disc and half ring. Also a harmonic Dirichlet problem was investigated in a quarter ring domain [3].…”

“…In this article, we extend some results to a general domain. Firstly we shall give a harmonic Green function based on conformal mapping in a ring sector with angle θ = π α , α ≥ 1 2 (when α = 1 or α = 2, it is the cases in [2,3] respectively), and then discuss a related Dirichlet problem for the Poisson equation explicitly.…”