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Nonlinear Beltrami equation: lower estimates of Schwarz lemma’s type
- Part of
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- Journal:
- Canadian Mathematical Bulletin / Volume 67 / Issue 3 / September 2024
- Published online by Cambridge University Press:
- 29 November 2023, pp. 533-543
- Print publication:
- September 2024
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- Article
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We study a nonlinear Beltrami equation
$f_\theta =\sigma \,|f_r|^m f_r$ in polar coordinates 
$(r,\theta ),$ which becomes the classical Cauchy–Riemann system under 
$m=0$ and 
$\sigma =ir.$ Using the isoperimetric technique, various lower estimates for 
$|f(z)|/|z|, f(0)=0,$ as 
$z\to 0,$ are derived under appropriate integral conditions on complex/directional dilatations. The sharpness of the above bounds is illustrated by several examples.
Beltrami Equation with Coefficient in Sobolev and Besov Spaces
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- Journal:
- Canadian Journal of Mathematics / Volume 65 / Issue 6 / 01 December 2013
- Published online by Cambridge University Press:
- 20 November 2018, pp. 1217-1235
- Print publication:
- 01 December 2013
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Our goal in this work is to present some function spaces on the complex plane
$\mathbb{C},\,X(\mathbb{C})$, for which the quasiregular solutions of the Beltrami equation, 
$\bar{\partial }f(z)\,=\,\mu (z)\partial f(z)$, have first derivatives locally in 
$X(\mathbb{C})$, provided that the Beltrami coefficient 
$\mu $ belongs to 
$X(\mathbb{C})$.
