Hydrodynamic normalization conditions in the theory of degenerate Beltrami equations





BMO, bounded mean oscillation, FMO, finite mean oscillation, degenerate Beltrami equations, hydrodynamic normalization


We study the existence of normalized homeomorphic solutions for the degenerate Beltrami equation fz = μ(z )f  in the whole complex plane C , assuming that its measurable coefficient μ(z ), | μ(z ) |<1 a. e., has compact support and the degeneration of the equation is controlled by the tangential dilatation quotient KT μ (z , z0) . We show that if KT μ (z , z0) has bounded or finite mean oscillation dominants, or satisfies the Lehto type integral divergence condition, then the Beltrami equation admits a regular homeomorphic   W1,1loc solution f with the hydrodynamic normalization at infinity. We also give integral criteria of Calderon-Zygmund or Orlicz types for the existence of the normalized solutions in terms of KT μ (z , z0) and the maximal dilatation Kμ (z ) .


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How to Cite

Gutlyanskiĭ, V. ., Ryazanov, V. ., Sevost’yanov, E. ., & Yakubov, E. . (2023). Hydrodynamic normalization conditions in the theory of degenerate Beltrami equations. Reports of the National Academy of Sciences of Ukraine, (2), 10–17. https://doi.org/10.15407/dopovidi2023.02.010