Neumann boundary value problems in fan-shaped domains

Abstract

n this article we give the solvability conditions and the integral representations of the solutions of the Neumann boundary value problem for the Cauchy-Riemann operator and the Beltrami operator with constant coefficient in a disc sector with angle $\vartheta=\frac{\pi}{n},\,n\in\mathbb N$. Moreover, the Neumann problem for second-order operators with the Bitsadze/Laplace operator as the main part is studied. Classical results of complex analysis are used to obtain the expressions of the solvability conditions and the integral representations for the solutions explicitly.