Beltrami equations with degenerate on arcs

A. N. Kondrashov

Matematičeskaâ fizika i kompʹûternoe modelirovanie, no. 5 (2014), pp. 24-39.

Résumé

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 (*).

Détail

Keywords: degenerate Beltrami equation, Beltrami equation of variable type, folds, solution with singularity, associated equation.

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