Generalized Cauchy-Riemann equations and relevant PDE
V. Gutlyanskii, V. Ryazanov, A. Salimov, R. Salimov
TL;DR
The paper surveys and systematizes the connection between Beltrami equations in the complex plane and generalized Cauchy-Riemann equations in the real plane, clarifying links to $A$-harmonic equations via the Hodge operator. It develops a rigorous, pointwise bridge between the complex coefficient $\mu$ and the matrix coefficient $B$, enabling transfer of Beltrami theory to the generalized CR system $\nabla v = B \nabla u$ and its boundary-value problems. Existence, representation, and regularity results are established under a range of ellipticity and growth conditions (e.g., $K_{\mu_B}$, $K^T_{\mu_B}$, BMO/VMO, Orlicz), yielding solvability for Dirichlet, Hilbert, Neumann, Poincaré, and Riemann problems with both continuous and measurable data. The work also connects to hydrodynamics in anisotropic media via the $A$-harmonic framework and shows how solutions extend to the plane using the Hodge matrix ${\mathbb H}$, broadening the applicability to physical contexts and complex boundary behaviors.
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.