Schwarz problem for higher‐order complex partial differential equations in the upper half plane. Issue 6 (4th January 2019)

• Record Type: Journal Article
• Authors: Aksoy, Ümit; Begehr, Heinrich; Çelebi, A. Okay
• Is Part Of: Mathematische Nachrichten. Volume 292:Issue 6(2019)
• Page Start: 1183
• Page End: 1193
• Publication Date: 2019-01-04
• Subjects: complex partial differential equation -- polyanalytic -- Schwarz problem -- upper half plane -- 30E25 -- 30G20 -- 35G15 -- 35G30; Mathematics -- Periodicals; 510.5
• DOI: 10.1002/mana.201800028
• Languages: English
• ISSNs: 0025-584X

Abstract

Abstract: Linear and nonlinear elliptic complex partial differential equations of higher‐order are considered under Schwarz conditions in the upper‐half plane. Firstly, using the integral representations for the solutions of the inhomogeneous polyanalytic equation with Schwarz conditions, a class of integral operators is introduced together with some of their properties. Then, these operators are used to transform the problem for linear equations into singular integral equations. In the case of nonlinear equations such a transformation yields a system of integro‐differential equations. Existence of the solutions of the relevant boundary value problems for linear and nonlinear equations are discussed via Fredholm theory and fixed point theorems, respectively.