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<p>Here we give a survey of consequences from the theory of the Beltrami equations in the complex plane ℂ to generalized Cauchy-Riemann equations ∇<i>υ</i> = <i>B</i>∇<i>u</i> in the real plane ℝ<sup>2</sup> and clarify the relationships of the latter to the <i>A</i>−harmonic equation div <i>A</i> grad <i>u</i> = 0 with the matrix-valued coefficients <i>A</i>, which is one of the main equations in the potential theory of hydromechanics (fluid mechanics) in anisotropic and inhomogeneous media. The survey includes various types of results as theorems on the existence, representation, and regularity of their solutions, in particular, for the main boundary value problems of Hilbert, Dirichlet, Neumann, Poincaré, and Riemann.</p>

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On generalized Cauchy–Riemann equations for anisotropic and inhomogeneous media

  • 期刊论文

  • 发表日期：

    2025年2月1日

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  • Vladimir Gutlyanskii,
  • Vladimir Ryazanov,
  • Artem Salimov,
  • Ruslan Salimov

摘要

Here we give a survey of consequences from the theory of the Beltrami equations in the complex plane ℂ to generalized Cauchy-Riemann equations ∇υ = B∇u in the real plane ℝ2 and clarify the relationships of the latter to the A−harmonic equation div A grad u = 0 with the matrix-valued coefficients A, which is one of the main equations in the potential theory of hydromechanics (fluid mechanics) in anisotropic and inhomogeneous media. The survey includes various types of results as theorems on the existence, representation, and regularity of their solutions, in particular, for the main boundary value problems of Hilbert, Dirichlet, Neumann, Poincaré, and Riemann.