Mathematics

B. A. Wertheim

ON THE APPROXIMATE CONSTRUCTION OF CERTAIN QUASICONFORMAL MAPPINGS

(Presented by Academician M. A. Lavrent’ev, 15 X 1957)

We consider the problem of constructing a quasiconformal mapping of the disk (T,\ |z|<1), onto a domain (G) in the (w)-plane by means of a function

[
w=w(z)=u(x,y)+iv(x,y),
\tag{1}
]

satisfying the linear elliptic system

[
\begin{aligned}
a u_x+b u_y&=v_y,\
-bu_x+a u_y&=-v_x,
\end{aligned}
\qquad
a(x,y)>0,
\tag{2}
]

under the normalization conditions

[
w(0)=w_0=0,\qquad w(t_1)=w_1=u_1+iv_1,
\tag{3}
]

where (w_0\in G,\ w_1\in \Gamma,\ |t_1|=1). The curve (\Gamma) is the boundary of the domain (G) and is given by the equation

[
F(u,v)=0.
\tag{4}
]

It is assumed that (a), (b), and (F) are twice continuously differentiable functions of their arguments (the first two in the disk (\overline T)), and that the second derivatives of (F) in a neighborhood of the line (\Gamma) satisfy the Lipschitz condition

[
\left|F''{u^p v^q}(u_1,v_1)-F''(u_2,v_2)\right|\leq
]

[
\leq K\left(|u_1-u_2|+|v_1-v_2|\right),\qquad p+q=2,\quad p,q=0,1,2;
\tag{5}
]

[
|\operatorname{grad} F|\geq \lambda>0.
\tag{6}
]

Let, on the boundary, (w(t)=u(t)+iv(t)), (t=e^{i\varphi}). Introduce in the space (H_\mu) (of functions continuous in the sense of Hölder, with the usual norm; see ((^1))) the linear functionals (k) and (l):

[
k[u(t)]=u(0,0),\qquad l[v(t)]=v(0,0)
\tag{7}
]

and the operator (S) of passage to the conjugate function, analogous to the integral operator with the Hilbert kernel in the conformal case:

[
v(t)=Su(t),\qquad l[Su(t)]\equiv 0.
\tag{8}
]

The actual computation of (k) and (S) requires solving the Dirichlet problem for the elliptic equation of second order in (u) that follows from system (2). The functional (l) is computed analogously, and it is evident that (|k|=)

(\equiv |z| \equiv 1). For a general investigation of the operator (S), the results of I. N. Vekua ([2], problem A) are applicable.

The mapping problem under consideration reduces to determining the functions ({u(t), v(t)}), (t=e^{i\varphi}), from the system of equations

[
F[u(t),v(t)]=0,
\tag{9}
]

[
L \equiv v(t)-Su(t)=0,\qquad K[u(t)]=0,
\tag{10}
]

[
L_1 \equiv f_1u(t_1)+g_1v(t_1)+h_1=0
\tag{11}
]

(here in (11) (L_1(u,v)=0) is the exact or approximate equation of the normal to the curve (\Gamma) at the point (w_1)); in what follows

[
f_1=-F_v[u_0(t_1),v_0(t_1)],\qquad
g_1=F_u[u_0(t_1),v_0(t_1)],\qquad
h_1=-f_1u_1-g_1v_1.
\tag{12}
]

Moreover, the solution must satisfy the following conditions: a) as (\varphi) varies by (2\pi), the point (w=u(t)+iv(t)) must traverse the contour (\Gamma) once in the positive direction; b) since the normal (11) intersects the line (\Gamma), besides (w_1), at least at one more point (we denote by (w_2) the intersection point nearest to (w_1)), one more condition is needed which would single out from the solutions of the system (9)—(11) the one satisfying (3), for example: (|w(t_1)-w_1|<|w_2-w_1|). Conditions a) and b) are taken into account in choosing the initial approximation (see below).

To solve the system (9)—(11) we apply Newton’s method, developed for the approximate solution of nonlinear functional equations by L. V. Kantorovich ([3–4]). Without loss of generality, we shall assume that the initial approximations (u_0(t)) and (v_0(t)) satisfy equations (10); the general case reduces to this by replacing (u_{0,1}=u_0-K[u_0]), (v_{0,1}=v_0-L[u_0,v_0]).

For definiteness, let us write the equations corresponding to the modified Newton method. Suppose that the (n)-th approximation ({u_n(t),v_n(t)}), (n=0,1,2,\ldots), has been found. To determine the corrections

[
u_{n+1}-u_n=u',\qquad v_{n+1}-v_n=v'
\tag{13}
]

we arrive at the linear boundary-value problem for the system (2) (instead of (u',v') we again write (u,v)):

[
F_u[u_0(t),v_0(t)]u(t)+F_v'[u_0(t),v_0(t)]v(t)=-F[u_n(t),v_n(t)]
\tag{14}
]

with the additional conditions

[
u(z)+iv(z)\big|_{z=0}=0,
\tag{15}
]

[
f_1u(t_1)+g_1v(t_1)=-L_1[u_n(t_1),v_n(t_1)].
\tag{16}
]

Since the system (2), by means of the “affine” substitution (u_1=au,\ v_1=v-bu), is reduced to the Carleman system, it becomes possible, for the investigation of the boundary-value problem (14), to apply the theory developed by I. N. Vekua for the linear boundary-value problem A ([2]).

Lemma 1. The index of problem A corresponding to problem (14) for system (2) is equal to (n=1).

As is known, in this case the general solution contains (2n+1=3) arbitrary real constants, which here are determined by means of the three conditions (15)—(16).

Lemma 2. The homogeneous problem corresponding to (14), with condition (15), for (n=1) has one and only one independent solution, nowhere, except for (z=0), vanishing.

The proof follows from the proof of one proposition of the work ([2]) (p. 283) concerning the case (n=0). We denote the considered

in the lemma the solution by (W_0(z)), normalizing it by (|W_0(t_1)|=1), and by (W_1(z)) and (W_2(z))—some solutions of the homogeneous problem corresponding to (14), without condition (15), satisfying the conditions: (W_1(0)=1,\quad W_2(0)=i). The existence of these two solutions follows from Lemmas 1 and 2.

Lemma 3. Problem (14) with conditions (15)—(16) has one and only one solution for any right-hand sides.

One of the possible ways of actually determining the corrections is connected with solving, by the method of successive approximations, the integral equation (8.96) of paper ((^2)), which here is conveniently written in the form
[
[I-L_{(A_0)}]V(z)=\Phi(z),
]
where (L_{(A_0)}) is an integral operator depending on a certain function (A_0). From the results of ((^2)) it follows that there exists a bounded inverse operator
[
[I-L_{(A_0)}]^{-1},
]
mapping the space (H_\mu) (of functions Hölder-continuous in the circle (\overline T), with the usual norm) into itself.

Introduce the notation:
[
A(t)=\frac{1}{2a}\frac{\partial}{\partial \overline z}(a-ib);
]
[
\Omega(z)=-\frac{1}{\pi}\iint_T \frac{A(t)\,dT}{t-z}
=\rho(z)+i\omega_1(z);
]
[
\alpha=\frac{1}{a}\,[F_u(u_0,v_0)+bF_v(u_0,v_0)],\qquad
\beta=F_v(u_0,v_0);
]
[
[(\alpha(t)-i\beta(t))e^{\Omega(t)}]{|t|=1}
=t^{-1}M(t)e^{i\omega_2(t)};
]
[
p_k(z)=\frac{1}{2\pi i}\int}\omega_k(t)\frac{t+z}{t-z}\frac{dt}{t
=\omega_k^(z)+i\psi_k^(z);
]
[
A_{0,k}(z)=\overline A(z)e^{2i[\omega_k^(z)-\omega_k(z)]},\qquad k=1,2;
]
[
\bigl|[I-L_{(A_{0,k})}]^{-1}\bigr|\le B_k;
]
[
c(t)=-\frac{e^{-\psi_2^(t)}}{M(t)},\qquad
\gamma_n(t)=c(t)F[u_n(t),v_n(t)];
]
[
N_0=\max\left[
\frac{1}{\mu\pi}(1+2^{1+\mu}+3^\mu)+\frac{\pi}{1-\mu};
\frac{4^{1+\mu}}{\mu\pi}
\right]+\frac{2\pi^{\mu-1}}{\mu};
]
[
B_0=4(1+|W_0^F(z)|)(1+|W_1(z)|+|W_2(z)|)|c(t)|\times
]
[
\times\left(1+\left|\frac{1+ib(z)}{a(z)}\right|\right)
\left|e^{\Omega(z)-ip_2(z)}\right|B_1N_0+\frac{1}{\lambda}|W_0(z)|;
]
[
N_1=4e^{4|\rho(t)+\psi_1^*(t)|}|a(t)|B_2N_0+2|b(t)|;
]
[
\eta_0=\max{|F[u_0(t),v_0(t)]|,\ |L_1[u_0(t_1),v_0(t_1)]|};
]
[
K_0=K(|u_0(t)|+|v_0(t)|+2r)(1+N_1)^2+\max|F''_{u^pv^q}|,
]
[
p+q=2;\qquad p,q=0,1,2;
]
[
r=2B_0\eta_0\sqrt{1+N_1^2}.
]

We formulate the conditions for convergence of the approximations, following from Theorem 1 of paper ((^5)).

Let the initial approximation correspond to a closed curve without multiple points
[
\Gamma_0:\quad w=u_0(t)+iv_0(t).
]
Construct an annular neighbor-

the part (R) of the line (\Gamma_0) formed by the circle of radius (r), whose center runs along (\Gamma_0), and suppose that in (R) conditions (5)—(6) are satisfied.

Theorem. Suppose the inequality

[
h_0=B_0^2K\bar{\gamma}_0<\frac12 .
]

is satisfied. Then in the domain

[
|u(t)-u_0(t)|\leq 2B_0\bar{\gamma}_0,\qquad
|v(t)-v_0(t)|\leq 2N_1B_0\bar{\gamma}_0
]

there exists a solution ({u^(t),v^(t)}) of the system of equations (9)—(11), and the successive approximations of the modified Newton process converge to this solution, with

[
|u_n(t)-u^*(t)|\leq \frac{q^n}{1-q}B_0\bar{\gamma}_0,\qquad
q=1-\sqrt{1-2h_0}<1 .
]

For conditions a) and b) to hold it is sufficient that

[
r<\min\left{\min_{|t|=1}|u_0(t)+iv_0(t)|;\; |w_2-w_1|-|w_0(t_1)-w_1|\right}.
]

In particular, the indicated method is applicable to the approximate construction of conformal mappings. In this case, under the usual normalization conditions (w(0)=0,\ w'(0)>0), the process of successive approximations resulting from the solution of the Riemann—Hilbert boundary-value problem can be expressed explicitly as

[
u_{n+1}(t)+iv_{n+1}(t)=u_n(t)+iv_n(t)+
]

[
+\,te^{-ip_2(t)}{\gamma_n(t)+iS\gamma_n(t)-i(k[\gamma_n(t)])\tg k[\omega_2(t)]}
\tag{17}
]

(in the present case (a\equiv 1,\ b=\Omega=0), (S) is the singular integral operator with Hilbert kernel; by Gauss’s theorem the functional (k) reduces to averaging; here the condition (\cos k[\omega_2(t)]\ne 0) is necessary, similar to that which appears when Newton’s method is applied to the nonlinear singular equation of Theodorsen’s method; this condition is fulfilled for a sufficiently good initial approximation).

Another application of Newton’s method to the approximate construction of conformal mappings in connection with the method of conjugate trigonometric series was recently proposed by G. A. Nikolaeva ((^6)).

The author expresses deep gratitude to Prof. L. I. Volkovyskii for suggesting the topic and for supervising the work.

Perm Mining Institute

Received
6 IX 1957

REFERENCES

(^{1}) N. I. Muskhelishvili, Singular Integral Equations, Moscow—Leningrad, 1946.
(^{2}) I. N. Vekua, Matem. sbornik, 31 (73), No. 2, 217 (1952).
(^{3}) L. V. Kantorovich, Uspekhi Mat. Nauk, 3, issue 6, 849 (1948).
(^{4}) L. V. Kantorovich, Dokl. Akad. Nauk SSSR, 80, No. 6, 849 (1951).
(^{5}) B. A. Vertgeim, Dokl. Akad. Nauk SSSR, 110, No. 5, 719 (1956).
(^{6}) G. A. Nikolaeva, Dokl. Akad. Nauk SSSR, 110, No. 2, 180 (1956).