A.V. Bitsadze's observation on bianalytic functions and the Schwarz problem. Issue 8 (3rd August 2019)

• Record Type: Journal Article
• Authors: Aksoy, Ümit; Begehr, Heinrich; Çelebi, A. Okay
• Is Part Of: Complex variables and elliptic equations. Volume 64:Issue 8(2019)
• Page Start: 1257
• Page End: 1274
• Publication Date: 2019-08-03
• Subjects: A. Soldatov; Bitsadze equation -- polyanalytic function -- plane domains -- harmonic Green function -- Schwarz boundary value problem -- Dirichlet and Neumann conditions; 30E25 -- 30G20 -- 31A05 -- 35J40; Functions of complex variables -- Periodicals; Differential equations, Elliptic -- Periodicals; 515.905
• DOI: 10.1080/17476933.2018.1504039
• Languages: English
• ISSNs: 1747-6933

Abstract

ABSTRACT: According to an observation of A.V. Bitsadze from 1948 the Dirichlet problem for bianalytic functions is ill-posed. A natural boundary condition for the polyanalytic operator, however, is the Schwarz condition. An integral representation for the solutions in the unit disc to the inhomogeneous polyanalytic equation satisfying Schwarz boundary conditions is known. This representation is extended here to any simply connected plane domain having a harmonic Green function. Some other boundary value problems are investigated with some Dirichlet and Neumann conditions illuminating that just the Schwarz problem is a natural boundary condition for the Bitsadze operator.