On Beltrami equations with a different-type of degeneracy on an arc

Suppose that, in a simply-connected domain $D\subset{\Bbb C}$, we are given the Beltrami equation (see [2, Chapter 2])

$$f_{\overline{z}}(z)=\mu(z)f_{z}(z). (*)$$

We will call case of the Beltrami equation with $|\mu(z)| < 1$ a.e. in $D$ by classical. The cases $|\mu(z)| < 1$ a.e. in $D$ and $|\mu(z)| > 1$ a.e. in $D$ differ in that, in the first case homeomorphisms do not change sense, and in the second they do. The difference is but formal here. Of interest is the situation when there simultaneously exist subdomains in $D$ in which $|\mu(z)| < 1$ a.e. and subdomains $D$ in which $|\mu(z)| > 1$ a.e. In this case the Beltrami equation is said to be alternating. The problem of the study of alternating Beltrami equations was posed by Volkovyskiǐ [3], and successful progress in this direction was made in [16;18]. Its solutions are described by mappings with folds, cusps, etc.
Assign to (*) the classical Beltrami equation with complex dilation
$$ \mu^{*}(z)=\left\{
\begin{array}{lcl} \mu(z)& |\mu(z)|\leq1,\\ 1 / \overline{\mu}(z)& |\mu(z)|>1. \end{array}
\right. $$
Below we call this equation associated with (*).
Suppose that there exists a Jordan arc $E\subset D$ dividing the domain $D$ into two simply-connected subdomains $D_1$ and $D_2$. Suppose also that (1) degenerates on $E$ and the nature of the degeneration is described by the following conditions:
\noindent(B1) The representation
\begin{eqnarray}|\mu(z)|=1+M(z)\delta (H(z)),\nonumber \end{eqnarray}
holds, where $M(z)$ is a measurable a.e. finite function in $D$; $\delta (t)$ is a continuous function such that $\delta (t)>0$ for $t\ne 0$ and $\delta (0)=0$; $H(z)\in C(D)\cap W^{1,2}_{\mathrm{loc}}(D)$, and also $\nabla H(z)\ne 0$ a.e. in $D$ and $H(z)<0$ in $D_1$, $H(z)>0$ in $D_2$.
\noindent(B2) there exists a continuous function $Z(z)\in W^{1,2}_{\mathrm{loc}}(D)$
$$ J(z)=H(z)+i Z(z)\in C(D)\cap W^{1,2}_{\mathrm{loc}}(D) $$
is a sense-preserving locally quasiconformal homeomorphism of $D$ onto $J(D)$.
Obviously, (B1) implies that $H(z)=0$ is the equation of $ E$. In what follows, we suppose that $I_1(z)=H_{x_1}Z_{x_2}-H_{x_2}Z_{x_1}$ is the Jacobian of $J(z)$, while $p_{J}(z)$ is its first Lavrent'ev characteristic, and $Q_J(D')=\mathop{\mathrm{ess\,sup}}_{D'}p_{J}(z)\geq1$. Then, since $J(z)$ is quasiconformal in $D'$, a.e. we have
\begin{equation}\nonumber |\nabla H(z)|^2+|\nabla Z(z)|^2\leq2Q_J(D')I_1(z)\leq2Q_J(D')|\nabla H(z)||\nabla Z(z)|. \end{equation}

Throughout the sequel, given an arbitrary real function $f(z)$, having gradient at a point $z\in D$, we put $\nabla f(z)=f_{x_1}+i f_{x_2}$ and $\overline{\nabla f(z)}=f_{x_1}-i f_{x_2}.$ Also we put
$$ S(z)=\left\lbrace
\begin{array}{ccc} \frac{\nabla H}{\overline{\nabla H}} & & z\in D_1, \\[15pt] \frac{\nabla Z}{\overline{\nabla Z}}& & z\in D_2, \end{array}
\right. $$

\begin{equation} \mathcal{F}_{\delta^{*}}(z)=f_{\delta^{*}}(x_1)+i x_2 \ \ \ \text{where} \ \ f_{\delta^{*}}(t)=\int_0^t{\delta^{*}} (\tau)d\tau. \nonumber \end{equation}

The main result of the article is as follows.
Theorem. Suppose that (B1) and (B1) are fulfilled, while for every subdomain $D'\Subset D$ there is a function $K(z)\in W^{1,2}(D')$ such that
\begin{equation} \iint\limits_{D'}\frac{ |\nabla K(z)|^2}{\delta(H)}dx_1dx_2<+\infty, \nonumber \end{equation}
and
\begin{equation} \nonumber \frac{1}{|M(z)|\delta^2(H)}\left|\mu(z)-S(z)\right|^2+\frac{1}{|M(z)|}\leq K(z). \end{equation}
for a.e. $z\in D'$. Put $T(z)=\mathcal{F}_{\delta^{*}}(J(z))$. Then there exists a homeomorphism $w=f(z):D\to f(D)\subset{\Bbb C}$ such that
(i) $f(z)$ is a solution with singularity E to the equation associated with (*);
(ii) $f(z)\in T^{*}W^{1,2}_{\mathrm{loc}}(D)$, $f^{-1}(w)\in W^{1,2}_{\mathrm{loc}}(f(D\setminus E))$, and, in the representation
\begin{equation} \nonumber f(z)=\varphi(T(z))=\varphi(\mathcal{F}_{\delta^{*}}(J(z))) \end{equation}
the mapping $\varphi$ has $W^{1,2}_{\mathrm{loc}}$- majorized first characteristic.
This homeomorphic solution with singularity $E$ is unique $T^{*}W^{1,2}_{\mathrm{loc}}(D)$ up to composition with a conformal mapping.



This result is a two-sided analog of Theorems 3, 4 of the paper [6].