abstract 78

Bulletin of Computational Applied Mathematics (Bull CompAMa) 

78

On boundary value problems for nonlinear systems of partial differential equations in the plane

Georgi Manjavidze, Wolfgang Tutschke

We present a joint work of Prof. G. Manjavidze and Prof. W. Tutschke, translated and edited by Profs. G. Akhalaia and N. Manjavidze, and which was completed by Prof. W. Tutschke in 2017. He added a new chapter 9 - "Outlook to recent trends" and which we were planning to publish. In this paper boundary value problems for the system of differential equations $$\frac{\partial w}{\partial \bar{z}}=F(z,w, \frac{\partial w}{\partial z})$$ in a bounded domain on the plane of the complex variable z the boundary of which consists of one or finite number of simple closed Liapunov curves are studied. Among them are the Modified Dirichlet problem for multiply connected domains, the Riemann-Hilbert problem as for simply connected domains as in holomorphic case. The existence of the solutions of above mentioned boundary value problems on the basis of a Schauder Principle are proved. The last chapter is completely devoted to recent investigations and open problems.

Keywords: Modified Dirichlet Problem; Riemann-Hilbert Problem; Schauder Principle; monogenic functions; Related fixed-point problem; Clifford-analytic normal form.

Cite this paper:

Manjavidze G., Tutschke W.

On boundary value problems for nonlinear systems of partial differential equations in the plane.

Bull. Comput. Appl. Math. (Bull CompAMa)

Vol. 11, No.1 pp.9-49, 2023.