On the Dirichlet problem for generalized Cauchy-Riemann equations

Abstract

Here we give a survey of consequences of the theory of the Beltrami equations from the complex analysis for the Dirichlet problem to generalized Cauchy-Riemann equation ∇v = B ∇u in the real plane R2 that describe flows of fluids in anisotropic and inhomogeneous media, where Bis a 2 × 2 matrix valued coefficient and the gradients ∇u and ∇v are interpreted as vector columns. Moreover, we clarify the relationships of the latter to the A-harmonic equation div (A∇u) = 0 with matrix valued coefficients A that is one of the main equations of the potential theory, namely, of the hydromechanics (fluid mechanics) in anisotropic and inhomogeneous media in the plane. The survey includes a series of effective integral criteria for existence of regular solutions of the Dirichlet problem with continuous data in arbitrary bounded simple connected domains to generalized Cauchy-Riemann equations with matrix coefficients in the case of anisotropic and inhomogeneous media.