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

A. N. Kondrashov

Matematičeskaâ fizika i kompʹûternoe modelirovanie, Tome 20 (2017) no. 5, pp. 5-16.

Résumé

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].

Détail

Keywords: first Lavrent'evs characteristic, degenerate Beltrami equation, alternating Beltrami equation, solution with singularity, associated equation.

Bibliographie

[1] P. P. Belinskiy, General Properties of Quasiconformal Mappings, Nauka. Sib. otd-nie Publ., Novosibirsk, 1974, 100 pp. | MR

[2] I. N. Vekua, Generalized analytic functions, Nauka Publ., Moscow, 1988, 512 pp. | MR

[3] L. I. Volkovyskiǐ, “Some problems of the theory of quasiconformal mappings”, Some Problems of Mathematics and Mechanics, Nauka Publ., Leningrad, 1970, 128–134 | MR

[4] V. M. Goldshteyn, Yu. G. Reshetnyak, Quasiconformal Mappings and Sobolev Spaces, Nauka Publ., Moscow, 1983, 284 pp. | MR

[5] A. N. Kondrashov, “Isothermic coordinates on sewing surfaces”, Science Journal of Volgograd State University. Mathematics. Physics, 6:37 (2016), 70–80

[6] A. N. Kondrashov, “On the theory of degenerate alternating beltrami equations”, Siberian Mathematical Journal, 53:6 (2012), 1321–1337 | MR

[7] A. N. Kondrashov, “On the theory of alternating Beltrami equation with many folds”, Science Journal of Volgograd State University. Mathematics. Physics, 2:19 (2013), 26–35

[8] A. N. Kondrashov, “Beltrami equations with degenerate on arcs”, Science Journal of Volgograd State University. Mathematics. Physics, 5:24 (2014), 24–39

[9] A. N. Kondrashov, “Alternating beltrami equation and conformal multifolds”, Science Journal of Volgograd State University. Mathematics. Physics, 5:30 (2015), 6–24

[10] V. G. Mazya, Sobolev Spaces, Izd-vo LGU Publ., Leningrad, 1985, 416 pp. | MR

[11] V. M. Miklyukov V.M., “Isothermic coordinates on singular surfaces”, Sbornik: Mathematics, 195:1 (2004), 69–88 | DOI | MR

[12] E. Kh. Yakubov, “Solutions of Beltrami's equation with degeneration”, Doklady Mathematics, 243:5 (1978), 1148–1149 | MR

[13] V. Gutlyanskii, V. Ryazanov, U. Srebro, E. Yakubov, The Beltrami Equations: A Geometric Approach, Springer, New York, 2012, xiv+301 pp. | MR

[14] M. Lavrentieff, “Sur une classe de représentations continues”, Sbornik: Mathematics, 42:4 (1935), 407–424

[15] O. Martio, V. M. Miklyukov, “On existence and uniqueness of degenerate Beltrami equations”, Complex Variables, 2004, no. 49, 647–656 | MR

[16] U. Srebro, E. Yakubov, “Branched folded maps and alternating Beltrami equations”, Journal d'analyse mathematique, 1996, no. 70, 65–90 | DOI | MR

[17] U. Srebro, E. Yakubov, “$\mu$-Homeomorphisms”, Contemporary Mathematics AMS, 211, 1997, 473–479 | DOI | MR

[18] U. Srebro, E. Yakubov, “Uniformization of maps with folds”, Israel mathematical conference proceedings, 1997, no. 11, 229–232 | MR