To the theory of semi-linear Beltrami equations

TL;DR

This paper investigates semi-linear Beltrami equations, establishing existence and representation of solutions, and applies these findings to complex physical phenomena modeled by related semi-linear Poisson equations.

Contribution

It introduces new existence and representation results for semi-linear Beltrami equations without boundary conditions and extends these to boundary value problems with measurable data.

Findings

Existence of regular solutions without boundary conditions.

Representation of solutions via Vekua type equations.

Applications to physical models like diffusion and plasma states.

Abstract

The present paper is devoted to the study of semi-linear Beltrami equations which are closely relevant to the corresponding semi-linear Poisson type equations of mathematical physics on the plane in anisotropic and inhomogeneous media. In its first part, applying completely continuous ope\-ra\-tors by Ahlfors-Bers and Leray--Schauder approach, we prove existence of regular solutions of the semi-linear Beltrami equations with no boundary conditions. Moreover, here we derive their representation through solutions of the Vekua type equations and generalized analytic functions with sources. As consequences, it is given a series of applications of these results to semi-linear Poisson type equations and to the corresponding equations of mathematical physics describing such phenomena as diffusion with physical and chemical absorption, plasma states and stationary burning in anisotropic and inhomogeneous media. The second part of the paper contains existence, representation and regularity results for nonclassical solutions to the Hilbert (Dirichlet) boundary value problem for semi-linear Beltrami equations and to the Poincare (Neumann) boundary value problem for semi-linear Poisson type equations with arbitrary boundary data that are measurable with respect to logarithmic capacity.