Two basic boundary-value problems for the inhomogeneous Cauchy-Riemann equation in an infinite sector (Q2911044)

The Schwarz and Dirichlet boundary value problems for the inhomogeneous Cauchy-Riemann equations in an infinite sector is solved by a conformal mapping. Boundary behaviors of the solutions and corresponding integrals are studied at the corner point. The solutions and the conditions of solvability are explicitly written.NEWLINENEWLINE For completeness: Similar investigations for the more general \(\mathbb R\)-linear problem were preformed in [\textit{Yu. V. Obnosov}, Russ. Math. 48, No. 7, 50--59 (2004); translation from Izv. Vyssh. Uchebn. Zaved., Mat. 48, No. 7, 53--62 (2004; Zbl 1100.30034)] and [the reviewer and \textit{P. M. Adler}, J. Phys. A, Math. Gen. 39, No. 14, 3545--3560 (2006; Zbl 1086.76068)]. The general theory of singular integrals on piecewise smooth curves is presented, for instance, in [\textit{L. P. Castro, R. Duduchava} and \textit{F.-O. Speck}, in: Toeplitz matrices and singular integral equations. The Bernd Silbermann anniversary volume. Basel: Birkhäuser. Oper. Theory, Adv. Appl. 135, 107--144 (2002; Zbl 1023.45002)]