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Vestnik Volgogradskogo gosudarstvennogo universiteta. Seriya 1. Mathematica. Physica, 2014, Issue 5(24), Pages 24–39
(Mi vvgum20)
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This article is cited in 3 scientific papers (total in 3 papers)
Mathematics
Beltrami equations with degenerate on arcs
A. N. Kondrashov Volgograd State University
Abstract:
In the recent paper [5] were obtained some conditions for the existence and uniqueness of solutions with singularity of the associated equation with the Beltrami equation
\begin{equation}
f_{\overline{z}}(z)=\mu(z)f_{z}(z).\tag{*}
\end{equation}
Here we gave geometric interpretation this results.
The main results are as follows.
Let $D\subset\mathbb{C}$ be a simply connected domain divided
by curve $E\subset D$ of class ${\rm C}^{(3)}$ into two subdomain $D_1$, $D_2$.
Theorem 1.
Suppose that $\mu(z)$ can be represented in the form
$$
\mu(z)=(1+\tilde{M}(z)\rho(d_E (z)))\mathop{\mathrm{e}^{2i\theta(z)}},
$$
where: 1) function $\rho (t)$ is continuous on $[0, +\infty)$, and $\rho(0)=0$ and $\rho(t)>0$ for $t\ne0$;
2) function $\theta(z)\in{\rm C}^{(1)}(D)$ is such that everywhere on $E$
$$
dz +\mathop{\mathrm{e}^{2i\theta (z)}}\overline{dz}\ne 0,
\label{Kondriuslovichsl}
$$
at $dz$ tangential to $E$;
3) complex-valued function $\tilde{M}(z)$ is measurable and almost everywhere in $D$
$$
\frac{1}{R}\leq|\mathrm{Re}\,\tilde{M}(z)|\leq R, \ |\mathrm{Im} \, \tilde{M}(z)|\leq R
\ \ (R\equiv\mathrm{const}).
$$
Then in some neighborhood of $O(E)$, there exists an solution with a singularity $E$ of the equation associated with the (*).
Theorem 2.
Suppose that 1) $H(z)\in{\rm C}^{1}(D)$, $\nabla H(z)\ne 0$ in $D$ and $E$
defined by the equation $H(z)=0$;
2) the function $\mu(z)$ can be written as:
$$
\mu(z)=\frac{\nabla H}{\overline{\nabla H}}+M^{*}(z)\rho(|H(z)|),
$$
where: 1) the function $\rho(t)$ is continuous on $[0, +\infty)$;
2) $\rho(0)=0$ and $\rho(t)>0$ for $t\ne0$, and
$\frac{1}{\rho(t)}$ has an integrable singularity
at zero;
3) $M^{*}(z)$ is complex-valued measurable function in $D$, and
$$
\Bigl|\mathrm{Re} \,\Bigl(\frac{\overline{\nabla H(z)}}{\nabla H(z)}
M^{*}(z)\Bigr)\Bigr|\geq\frac{1}{C_1},
\ \ |M^{*}(z)|\leq C_2 \ \ (C_1, \; C_2\equiv\mathrm{const}).
$$
Then in some neighborhood of $O(E)$, there exists an solution with a singularity $E$ of the equation associated with the (*).
Corollary.
Assume that
$$
\mu(z)=\frac{\nabla d_E (z)}{\overline{\nabla d_E (z)}}+M^{*}(z)
\rho(d_E(z)),
$$
where: 1) the function $\rho(t)$ is continuous on $[0, +\infty)$;
2) $\rho(0)=0$ and $\rho(t)>0$ for $t\ne0$, and
$\frac{1}{\rho(t)}$ has an integrable singularity
at zero;
3) $M^{*}(z)$ is complex-valued measurable function in $D$, and
$$
\Bigl|\mathrm{Re} \, \Bigl(
\frac{\overline{\nabla d_E(z)}}{\nabla d_E(z)}M^{*}(z)\Bigr)
\Bigr|\geq\frac{1}{C_1}, \ \ |M^{*}(z)|\leq C_2 \ (C_1,\; C_2\equiv\mathrm{const}).\nonumber
$$
Then in some neighborhood of $O(E)$, there exists an solution with a singularity $E$ of the equation associated with the (*).
Keywords:
degenerate Beltrami equation, Beltrami equation of variable type, folds, solution with singularity, associated equation.
Citation:
A. N. Kondrashov, “Beltrami equations with degenerate on arcs”, Vestnik Volgogradskogo gosudarstvennogo universiteta. Seriya 1. Mathematica. Physica, 2014, no. 5(24), 24–39
Linking options:
https://www.mathnet.ru/eng/vvgum20 https://www.mathnet.ru/eng/vvgum/y2014/i5/p24
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