Harmonic Dirichlet Problem in a Ring Sector

Abstract

In this paper, we construct a harmonic Green function by reflection method in a general ring sector with angle θ = π α and α ≥ 1 2 , then the related harmonic Dirichlet problem for the Poisson equation is discussed explicitly.

Schwarz boundary value problems for polyanalytic equation in a sector ring

“…The theory of the classical boundary value problems for analytic functions has been applied directly or indirectly in many different fields [1][2][3], such as signal analysis, crack and elasticity, orthogonal polynomials, time-frequency and so on. This makes a great interest in the investigation of boundary value problems for complex partial differential equations in different domains [4][5][6][7][8][9][10][11][12][13][14][15][16][17][18][19][20]. Especially, in [16,17], four basic boundary value problems for the Cauchy-Riemann equation and the Neumann problem for Bitsadze equation were studied in a ring domain by constructing kernel functions and integral expression formulas, respectively.…”

“…In [18], some harmonic boundary value problems for the Poisson equation were investigated in upper half ring domain with the help of building a modified Cauchy-Pompeiu formula, harmonic Green and Neumann functions. Dirichlet and Neumann problems for the Poisson equation in a quarter sector ring and the general sector ring were respectively solved in [19,20] through establishing proper Poisson kernels on basis of reflection principle.…”