MATHEMATICS

V. S. VINOGRADOV

ON SOME BOUNDARY-VALUE PROBLEMS FOR QUASILINEAR ELLIPTIC SYSTEMS OF FIRST ORDER IN THE PLANE

(Presented by Academician I. M. Vinogradov on 6 III 1958)

We shall write the quasilinear system

[
\begin{aligned}
&a_{11}(x,y,u,v)u_x+a_{12}(x,y,u,v)u_y+\
&\quad +b_{11}(x,y,u,v)v_x+b_{12}(x,y,u,v)v_y+d_1(x,y,u,v)=0,\
&a_{21}(x,y,u,v)u_x+a_{22}(x,y,u,v)u_y+\
&\quad +b_{21}(x,y,u,v)v_x+b_{22}(x,y,u,v)v_y+d_2(x,y,u,v)=0
\end{aligned}
]

in the complex form more convenient for us

[
\frac{\partial w}{\partial \bar z}
+\mu_1(z,w)\frac{\partial w}{\partial z}
+\mu_2(z,w)\frac{\partial \bar w}{\partial \bar z}
+d(z,w)=0,
]

[
z=x+iy,\qquad w=u+iv,
\tag{1}
]

[
\frac{\partial}{\partial \bar z}
=\frac12\left(\frac{\partial}{\partial x}
+i\frac{\partial}{\partial y}\right),\qquad
\frac{\partial}{\partial z}
=\frac12\left(\frac{\partial}{\partial x}
-i\frac{\partial}{\partial y}\right).
]

Here (\mu_1) and (\mu_2) are certain rational functions of (a_{ij}, b_{ij}); (d(z,w)) is a rational function of (a_{ij}, b_{ij}, d_i).

The coefficients of our system are defined when (z) belongs to a certain simply connected domain (G), whose boundary (\Gamma) has continuous curvature, and for all (w) in the complex plane (E).

We shall assume that system (1) is uniformly elliptic with respect to all (w); in our notation this condition is written in the form

[
|\mu_1(z,w)|+|\mu_2(z,w)|\leq \mu_0<1.
\tag{2}
]

Concerning the coefficients we make the following assumptions:

1. (\mu_i(z,w)), as functions of (z) for fixed (w), are measurable, and with respect to (w) satisfy the Lipschitz condition

[
|\mu_i(z,w_1)-\mu_i(z,w_2)|\leq K|w_1-w_2|.
\tag{3}
]

1. (d(z,w)), as a function of (z) for fixed (w), belongs to (L_p(G)), and with respect to (w) satisfies the condition

[
d(z,w)=d_0(z,w)+d_1(z,w)w+d_2(z,w)\bar w,
\tag{4}
]

where (|d_i(z,w)|_{L_p}2).

the boundary-value problem

[
\alpha u+\beta v\bigm|_{\Gamma}=0, \tag{5}
]

where (\alpha(t), \beta(t)) are Hölder-continuous on (\Gamma), and (\alpha^2+\beta^2=1), for the system (1) can always be reduced to the case when the domain (G) is the circle (|z|\leq 1), and the boundary condition is written in the form ({}^{1})

[
\operatorname{Re}{z^{-n}w(z)}\bigm|_{\Gamma}=0,
]

where (n) is the index of the problem; it is equal to the change of (\arg{\alpha(t)+i\beta(t)}) under one circuit of the point (t) around the contour (\Gamma) counterclockwise ({}^{2}).

We shall consider problem (1)—(5) in the following formulations.

The case of nonnegative index (n\geq 0).

Problem 1. Find a function (w(z)\in W_p^{(1)}(G)), (p>2), which satisfies equation (1) and the boundary conditions

[
\operatorname{Re}{z^{-n}w(z)}\bigm|{\Gamma}=0,\qquad
\int} z^{-k}w(z)\,ds=0\quad (k=0,1,\ldots,2n). \tag{6
]

The case of negative index (n<0).

Problem 2. Find a function (w(z)\in W_p^{(1)}(G)), (p>2), which satisfies equation (1), and (2|n|-1) real constants (\lambda_0,\lambda_{\pm1},\ldots,\lambda_{\pm|n|-1}) such that on the boundary the condition

[
\operatorname{Re}{z^{-n}w(z)}\bigm|{\Gamma}
=
\operatorname{Re}\left{\lambda_0+
\sum}^{|n|-1}(\lambda_k+i\lambda_{-k})z^k\right}\biggm|_{\Gamma
\tag{7}
]

is fulfilled.

Analogously to how this was done for linear systems in ({}^{1,3}), our problems reduce to an equivalent nonlinear singular integral equation

[
\rho+\mu_1(z,T_n\rho)S_n\rho+\mu_2(z,T_n\rho)\overline{S_n\rho}
+d(z,T_n\rho)=0, \tag{8}
]

where

[
T_n\rho=-\frac{1}{\pi}\iint_G
\left[\frac{\rho(\zeta)}{\zeta-z}+
\frac{z^{2n+1}\rho(\bar{\zeta})}{1-z\zeta}\right]\,dT_\zeta
\qquad \text{for } n\geq 0; \tag{9}
]

[
T_n\rho=-\frac{1}{\pi}\iint_G
\left[\frac{\rho(\zeta)}{\zeta-z}+
\frac{\bar{\zeta}^{\,2|n|-1}\rho(\bar{\zeta})}{1-z\bar{\zeta}}\right]\,dT_\zeta
\qquad \text{for } n<0;
]

[
S_n\rho=\frac{\partial}{\partial z}T_n\rho.
]

Let (\omega) be an arbitrary function from (L_p(G)); consider the linear equation

[
\rho+\mu_1(z,T_n\omega)S_n\rho+\mu_2(z,T_n\omega)\overline{S_n\rho}
]

[
{}+d_1(z,T_n\omega)T_n\rho+d_2(z,T_n\omega)\overline{T_n\rho}
+d_0(z,T_n\omega)=0. \tag{10}
]

The solution of this equation exists and is unique; denote it by (\rho=M\omega) ({}^{1}). (M\omega) is a nonlinear operator.

Lemma 1. The operator (M\omega) maps the whole space (L_p(G)) into a certain ball of finite radius; the latter depends only on (\mu_0, K_1).

Proof. In view of conditions (2), (4), uniform with respect to (w), the lemma is a consequence of Theorem 1 of ({}^{3}).

Lemma 2. The operator (M\omega) takes a weakly convergent sequence into a weakly convergent one, i.e. the operator (M\omega) is weakly continuous.

Proof. Let (\omega_m) converge weakly to (\omega_0). Then (\rho_m=M\omega)

there is a solution of equation (10) for (\omega=\omega_m). In view of the weak convergence of (\omega_m), (|\omega_m|_{L_p}\leq K_0). But (T_n\omega_m) converges to (T_n\omega_0) uniformly in (G), by virtue of the complete continuity of the embedding operator (W_p^{(1)}(G)), (p>2), in (C(D)).

By Lemma 1 and the properties of the operators (T_np) and (S_np), the quantities (|\rho_m|{L_p}), (|S_n\rho_m|), (|T_n\rho_m|) are bounded in the aggregate.

The difference ((\rho_m-\rho_0)) satisfies the equation

[
\begin{aligned}
&(\rho_m-\rho_0)+\mu_1(z,T_n\omega_0)S_n(\rho_m-\rho_0)
+\mu_2(z,T_n\omega_0)\overline{S}_n(\rho_m-\rho_0)+\
&\quad +d_1(z,T_n\omega_0)T_n(\rho_m-\rho_0)
+d_2(z,T_n\omega_0)\overline{T}_n(\rho_m-\rho_0)=\
&=[\mu_1(z,T_n\omega_0)-\mu_1(z,T_n\omega_m)]S_n\rho_m
+[\mu_2(z,T_n\omega_0)-\
&\quad -\mu_2(z,T_n\omega_m)]\overline{S}_n\rho_m
+[d_1(z,T_n\omega_0)-d_1(z,T_n\omega_m)]T_n\rho_m+\
&\quad +[d_2(z,T_n\omega_0)-d_2(z,T_n\omega_m)]\overline{T}_n\rho_m
+[d_0(z,T_n\omega_0)-d_0(z,T_n\omega_m)].
\end{aligned}
\tag{11}
]

In view of (3), (4) and what was said above, the right-hand side of equation (11) converges weakly to zero, while the inverse operator of the equation is linear, bounded, and its norm does not depend on (m). Therefore ((\rho_m-\rho_0)\to 0) weakly in (L_p(G)).

Remark. It can be proved that (M\omega_m) converges strongly in (L_p(G)) to (M\omega_0).

Theorem 1. There exists at least one solution of Problem 1 (Problem 2).

Proof. For the solvability of Problem 1 (Problem 2) it is sufficient that the operator (M\omega) have a fixed point. By Lemmas 1 and 2, applying to the operator (M\omega) Schauder’s principle ((^4)) for weakly continuous mappings of functional spaces (the sphere in (L_p(G)) is weakly compact), we obtain the existence of a solution of equation (9) and, consequently, of equation (8) and of our problems.

Theorem 2. The solution of Problem 1 (Problem 2) is unique.

Proof. Suppose that there exist two solutions (\rho_1,\rho_2) of Problem 1 (Problem 2); then equation (8) has two solutions. Therefore the difference (\rho_2-\rho_1), by virtue of conditions (3), (4), satisfies the linear homogeneous equation

[
\begin{aligned}
&(\rho_2-\rho_1)+\mu_1(z,T_n\rho_2)S_n(\rho_2-\rho_1)
+\mu_2(z,T_n\rho_2)\overline{S}_n(\rho_2-\rho_1)+\
&\quad +\left{
\frac{d(z,T_n\rho_2)-d(z,T_n\rho_1)}
{T_n\rho_2-T_n\rho_1}
+\frac{\mu_1(z,T_n\rho_2)-\mu_1(z,T_n\rho_1)}
{T_n\rho_2-T_n\rho_1}S_n\rho_1+\right.\
&\quad \left.
+\frac{\mu_2(z,T_n\rho_2)-\mu_2(z,T_n\rho_1)}
{T_n\rho_2-T_n\rho_1}\overline{S}_n\rho_1
\right}T_n(\rho_2-\rho_1)=0 .
\end{aligned}
]

By the uniqueness theorem for our equation (1), (\rho_2-\rho_1=0). For the case when (\mu_2(z,w)=0), this problem was considered by Nitsche ((^5)). In conclusion I express my gratitude to I. N. Vekua, under whose direction this work was carried out.

Mathematical Institute named after V. A. Steklov
Academy of Sciences of the USSR

Received
3 III 1958

CITED LITERATURE

(^1) V. S. Vinogradov, DAN, 118, No. 6 (1958).
(^2) I. N. Vekua, Matem. sborn., 31 (73), No. 2 (1952).
(^3) V. S. Vinogradov, DAN, 121, No. 3 (1958).
(^4) J. Schauder, Math. Zs., 26, 417 (1927).
(^5) I. Nitsche, Math. Nachr., 14, 3 (1955).