AV Bitsadze's observation on bianalytic functions and the Schwarz problem


AKSOY Ü., Begehr H., ÇELEBİ A. O.

COMPLEX VARIABLES AND ELLIPTIC EQUATIONS, vol.64, no.8, pp.1257-1274, 2019 (Peer-Reviewed Journal) identifier identifier

  • Publication Type: Article / Article
  • Volume: 64 Issue: 8
  • Publication Date: 2019
  • Doi Number: 10.1080/17476933.2018.1504039
  • Journal Name: COMPLEX VARIABLES AND ELLIPTIC EQUATIONS
  • Journal Indexes: Science Citation Index Expanded, Scopus
  • Page Numbers: pp.1257-1274
  • Keywords: A, Soldatov, Bitsadze equation, polyanalytic function, plane domains, harmonic Green function, Schwarz boundary value problem, Dirichlet and Neumann conditions

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.