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Mat. Zametki, 2011, Volume 90, Issue 3, Pages 445–453 (Mi mz8406)  

On Quasilinear Beltrami-Type Equations with Degeneration

E. A. Sevost'yanov

Institute of Applied Mathematics and Mechanics, Ukraine National Academy of Sciences, Donetsk

Abstract: We consider the solvability problem for the equation $f_{\overline{z}}=\nu(z,f(z)) f_z$, where the function $\nu(z,w)$ of two variables can be close to unity. Such equations are called quasilinear Beltrami-type equations with ellipticity degeneration. We prove that, under some rather general conditions on $\nu(z,w)$, the above equation has a regular homeomorphic solution in the Sobolev class $W_{\operatorname{loc}}^{1,1}$. Moreover, such solutions $f$ satisfy the inclusion $f^{ -1}\in W_{\operatorname{loc}}^{1,2}$.

Keywords: quasilinear Beltrami-type equation, existence theorem, regular homeomorphic solution, Sobolev class, homeomorphism, Carathéodory condition, function of bounded mean oscillation

DOI: https://doi.org/10.4213/mzm8406

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English version:
Mathematical Notes, 2011, 90:3, 431–438

Bibliographic databases:

UDC: 517.5
Received: 07.03.2009
Revised: 03.07.2010

Citation: E. A. Sevost'yanov, “On Quasilinear Beltrami-Type Equations with Degeneration”, Mat. Zametki, 90:3 (2011), 445–453; Math. Notes, 90:3 (2011), 431–438

Citation in format AMSBIB
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