Generalized Cauchy-Riemann equations and relevant PDE

TL;DR

This paper surveys the connections between generalized Cauchy-Riemann equations, Beltrami equations, and A-harmonic equations, highlighting their implications for potential theory and boundary value problems in complex and real planes.

Contribution

It clarifies the relationships between generalized Cauchy-Riemann equations and A-harmonic equations, and reviews existence, representation, and regularity results for their solutions.

Findings

Relationships between Beltrami and generalized Cauchy-Riemann equations clarified

Existence and regularity theorems for solutions discussed

Applications to boundary value problems in potential theory

Abstract

Here we give a survey of consequences from the theory of the Beltrami equations in the complex plane $\mathbb C$ to generalized Cauchy-Riemann equations $\nabla v = B \nabla u$ in the real plane $\mathbb R^2$ and clarify the relationships of the latter to the $A-$harmonic equation ${\rm div} A\,{\rm grad}\, u = 0$ with matrix valued coefficients $A$ that is one of the main equations of the potential theory, namely, of the hydro\-mechanics (fluid mechanics) in anisotropic and inhomogeneous media. The survey includes various types of results as theorems on existence, representation and regularity of their solutions, in particular, for the main boundary value problems of Hilbert, Dirichlet, Neumann, Poincare and Riemann.