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<p>A series of effective integral criteria for the existence of regular solutions of the Dirichlet problem with continuous data to general degenerate Beltrami equations <InlineEquation ID="IEq1"> <EquationSource Format="TEX">\(f_{\bar{z}}=\mu (z){f_z} + \nu (z)\overline{f_z}\)</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <msub> <mi>f</mi> <mover accent="true"> <mrow> <mi>z</mi> </mrow> <mrow> <mo stretchy="false">¯</mo> </mrow> </mover> </msub> <mo>=</mo> <mi>μ</mi> <mrow> <mo stretchy="false">(</mo> <mi>z</mi> <mo stretchy="false">)</mo> </mrow> <msub> <mi>f</mi> <mi>z</mi> </msub> <mo>+</mo> <mi>ν</mi> <mrow> <mo stretchy="false">(</mo> <mi>z</mi> <mo stretchy="false">)</mo> </mrow> <mover> <msub> <mi>f</mi> <mi>z</mi> </msub> <mo>¯</mo> </mover> </mrow> </math></EquationSource> </InlineEquation> in arbitrary bounded, simply connected domains <i>D</i> of the complex plane <InlineEquation ID="IEq2"> <EquationSource Format="TEX">\(\mathbb {C}\)</EquationSource> <EquationSource Format="MATHML"><math> <mi mathvariant="double-struck">C</mi> </math></EquationSource> </InlineEquation> have been established. The obtained results are based on the very deep theory of prime ends by Caratheodory, but our main theorems are formulated without using this term. They can be applied to hydromechanics (mechanics of fluids) in anisotropic and inhomogeneous media.</p>

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Dirichlet problem for general Beltrami equations in simply connected domains

  • 期刊论文

  • 发表日期：

    2026年3月24日

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  • V. Gutlyanskiĭ,
  • V. Ryazanov,
  • R. Salimov,
  • E. Sevost’yanov

摘要

A series of effective integral criteria for the existence of regular solutions of the Dirichlet problem with continuous data to general degenerate Beltrami equations \(f_{\bar{z}}=\mu (z){f_z} + \nu (z)\overline{f_z}\) f z ¯ = μ ( z ) f z + ν ( z ) f z ¯ in arbitrary bounded, simply connected domains D of the complex plane \(\mathbb {C}\) C have been established. The obtained results are based on the very deep theory of prime ends by Caratheodory, but our main theorems are formulated without using this term. They can be applied to hydromechanics (mechanics of fluids) in anisotropic and inhomogeneous media.