非线性贝尔特拉米方程：施瓦茨两难类型的下限估计值

Canadian Mathematical Bulletin Pub Date : 2023-11-29 DOI:10.4153/s0008439523000942

Igor Petkov, Ruslan Salimov, Mariia Stefanchuk

引用次数: 0

摘要

我们研究了极坐标 $(r,\theta ) $ 下的非线性贝尔特拉米方程 $f_\theta =\sigma \,|f_r|^m f_r$，该方程在 $m=0$ 和 $\sigma =ir 条件下成为经典的考奇-黎曼系统。利用等周技术，在复数/方向扩张的适当积分条件下，导出了当 $z\to 0 时 $|f(z)|/|z|, f(0)=0,$ 的各种下限估计。通过几个例子说明了上述边界的尖锐性。

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Nonlinear Beltrami equation: lower estimates of Schwarz lemma’s type

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.