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The Dirichlet problem for the Beltrami equations with sources

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Abstract

The paper is devoted to the study of the Dirichlet problem Re ω(z) → φ(ζ) as z → ζ, z ∈ D, ζ ∈ ∂D, with continuous boundary data φ : ∂D → ℝ for Beltrami equations \( {\omega}_{\overline{z}} \) = μ(z)ωz + σ(z), |μ(z)| < 1 a.e., with sources σ : D → ℂ in the case of locally uniform ellipticity. In this case, we have established a series of effective integral criteria of the BMO, FMO, Calderon-Zygmund, Lehto, and Orlicz types on the singularities of the equations at the boundary for the existence, representation, and regularity of solutions in arbitrary bounded domains D of the complex plane ℂ with no boundary component degenerated to a single point for sources σ in Lp (D), p > 2, with compact support in D. Moreover, we have proved the existence, representation, and regularity of weak solutions of the Dirichlet problem in such domains for the Poisson-type equation div[A(z)∇ u(z)] = g(z), whose source g ∈ Lp(D), p > 1, has compact support in D and whose matrix-valued coefficient A(z) guarantees its locally uniform ellipticity.

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Authors and Affiliations

  1. Institute of Applied Mathematics and Mechanics of the NAS of Ukraine, Sloviansk, Institute of Mathematics of the NAS of Ukraine, Kiev, Ukraine

    Vladimir Gutlyanskiĭ, Vladimir Ryazanov & Olga Nesmelova

  2. Rolon Institute of Technology, Rolon, Israel

    Eduard Yakubov

Authors
  1. Vladimir Gutlyanskiĭ
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  2. Vladimir Ryazanov
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  3. Olga Nesmelova
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  4. Eduard Yakubov
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Correspondence to Vladimir Gutlyanskiĭ.

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Translated from Ukrains’kiĭ Matematychnyĭ Visnyk, Vol. 20, No. 1, pp. 24-59, January-March, 2023.

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Gutlyanskiĭ, V., Ryazanov, V., Nesmelova, O. et al. The Dirichlet problem for the Beltrami equations with sources. J Math Sci 273, 351–376 (2023). https://doi.org/10.1007/s10958-023-06503-0

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