Generalized Cauchy-Riemann equations and relevant PDE

2024/11/13 by Vladimir Gutlyanskiî, Gutlyanskii, V., V. Ryazanov +5
Mathematics · #30C65 #30E25 Secondary 35J25 #76-02 #Algebraic and Geometric Analysis #Analysis of PDEs (math.AP) #Complex Variables (math.CV) #FOS: Mathematics #Primary 30C62

paper · pdf · doi:10.48550/arxiv.2411.08941

openalex publication_date 2024/11/13 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28

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 ∇ v = B ∇ u in the real plane \mathbb R² 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.

Related

• On boundary value problems for inhomogeneous Beltrami equations
• Toward the theory of Dirichlet problem for the degenerate Beltrami equations
• Beltrami equations and mappings with asymptotic homogeneity at infinity
• Fluid dynamics solutions obtained from the Riemann invariant approach
• Beltrami equation for the harmonic diffeomorphisms between surfaces