MATHEMATICS

V. S. VINOGRADOV

ON THE BOUNDEDNESS OF SOLUTIONS OF BOUNDARY VALUE PROBLEMS FOR LINEAR ELLIPTIC SYSTEMS OF FIRST ORDER IN THE PLANE

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

Consider the boundary value problem

\[ \alpha u+\beta v\bigm|_{L}=0 \]

for the system of equations of elliptic type

\[ \begin{aligned} a_{11}u_x+a_{12}u_y+b_{11}v_x+b_{12}v_y+c_{11}u+c_{12}v&=g_1,\\ a_{21}u_x+a_{22}u_y+b_{21}v_x+b_{22}v_y+c_{21}u+c_{22}v&=g_2 \end{aligned} \]

in some simply connected domain \(G\) with boundary \(L\); \(\alpha(t)\), \(\beta(t)\) are functions given on \(L\), Hölder-continuous; \(\alpha^2+\beta^2=1\). This problem can always be reduced to the following form \((^{1})\):

\[ \frac{\partial w}{\partial \bar z} +\mu_1(z)\frac{\partial w}{\partial z} +\mu_2(z)\frac{\partial \bar w}{\partial \bar z} +a_1(z)w+a_2(z)\bar w=g(z); \tag{1} \]

\[ \operatorname{Re}\{z^{-n}w(z)\}\bigm|_{\Gamma}=0; \tag{2} \]

\[ z=x+iy;\quad w=u+iv;\quad \frac{\partial}{\partial \bar z} =\frac12\left(\frac{\partial}{\partial x}+i\frac{\partial}{\partial y}\right); \quad \frac{\partial}{\partial z} =\frac12\left(\frac{\partial}{\partial x}-i\frac{\partial}{\partial y}\right); \]
\(\Gamma\) is the boundary of the disk \(D\) \((|z|\leqslant 1)\);
\[ n=\frac{1}{2\pi}[\arg(\alpha+i\beta)]_{L} \]
is the index of our problem \((^{*})\). We shall assume that \(\mu_1,\mu_2\) are measurable functions in the disk \(D\).

The ellipticity condition for the system written in the form (1) is given by the estimate

\[ |\mu_1(z)|+|\mu_2(z)|\leqslant \mu_0<1. \tag{3} \]

With respect to the remaining coefficients we shall assume:

\[ \begin{gathered} a_1(z),\ a_2(z),\ g(z)\in L_p(D),\quad p>2;\\ \|a_1\|_{L_p},\ \|a_2\|_{L_p},\ \|g\|_{L_p}\leqslant K. \end{gathered} \tag{4} \]

We shall investigate our problem in the following settings:

Case of nonnegative index \(n\geqslant 0\).

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

\[ \operatorname{Re}\{z^{-n}w(z)\}\bigm|_{\Gamma}=0,\qquad \int_{\Gamma} z^{-k}w(z)\,ds=0\quad (k=0,1,\ldots,2n). \tag{5} \]

The case of negative index \(n<0\).

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

\[ \left.\operatorname{Re}\{z^{-n}w(z)\}\right|_{\Gamma} = \operatorname{Re}\left\{\lambda_0+\sum_{k=1}^{|n|-1}(\lambda_k+i\lambda_{-k})z^k\right\}. \tag{6} \]

By the method that we used in [1], Problems 1 and 2 can be reduced to the equivalent singular integral equation

\[ \rho+\mu_1S_n\rho+\mu_2\overline{S}_n\rho+a_1T_n\rho+a_2\overline{T}_n\rho=g, \tag{7} \]

which is equivalent to an equation of Fredholm type, and whose corresponding homogeneous equation has no solutions different from zero [1].

We denoted:

for \(n\geq 0\),

\[ T_n\rho=-\frac{1}{\pi}\iint_D \left[ \frac{\rho(\zeta)}{\zeta-z} + \frac{z^{2n+1}\overline{\rho(\zeta)}}{1-z\overline{\zeta}} \right]\,dT_\zeta, \tag{8} \]

\[ S_n\rho=\frac{\partial}{\partial z}T_n\rho = -\frac{1}{\pi}\iint_D \left[ \frac{\rho(\zeta)}{(\zeta-z)^2} + \frac{z^{2n+1}\overline{\zeta}\,\rho(\overline{\zeta})}{(\zeta-z\overline{\zeta})^2} \right]\,dT_\zeta - \]

\[ -\frac{2n+1}{\pi}\iint_D \frac{z^{2n}\rho(\overline{\zeta})}{1-z\overline{\zeta}}\,dT_\zeta = S_n^0\rho+T_n^0\rho; \tag{9} \]

for \(n<0\),

\[ T_n\rho=-\frac{1}{\pi}\iint_D \left[ \frac{\rho(\zeta)}{\zeta-z} + \frac{\overline{\zeta}^{\,2|n|-1}\overline{\rho(\zeta)}}{1-z\overline{\zeta}} \right]\,dT_\zeta, \tag{10} \]

\[ S_n\rho=\frac{\partial}{\partial z}T_n\rho = -\frac{1}{\pi}\iint_D \left[ \frac{\rho(\zeta)}{(\zeta-z)^2} + \frac{\overline{\zeta}^{\,2|n|}\overline{\rho(\zeta)}}{(1-z\overline{\zeta})^2} \right]\,dT_\zeta. \tag{11} \]

The operator \(T_n^0\rho\) for \(n\geq0\) is completely continuous, and the norms of the operators \(S_n^0\rho\) for \(n\geq0\) and \(S_n\rho\) for \(n<0\) are equal to one if \(\rho\in L_2(D)\) [1].

Theorem 1. The solution of equation (7) is bounded in the norm in \(L_p(D)\) by a number depending only on \(\mu_0\) and \(K\).

Proof. Suppose the contrary. Let there exist a sequence of coefficients \(\mu_{1m}, \mu_{2m}, a_{1m}, a_{2m}, g_m\), satisfying conditions (3), (4), such that the sequence of equations

\[ \rho_m+\mu_{1m}S_n\rho_m+\mu_{2m}\overline{S}_n\rho+a_{1m}T_n\rho_m+a_{2m}\overline{T}\rho_m=g_m \tag{12} \]

tends to infinity in the norm of \(L_p(D)\).

Set \(\tau_m=\rho_m/\|\rho_m\|\), \(\|\tau_m\|=1\). In view of the weak compactness of \(\{\tau_m\}\), one can extract from it a weakly convergent subsequence. Let this be \(\tau_m\to\tau_0\). Set \(w_m=T_n\tau_m\).

Consider two cases: \(n\geq0\) and \(n<0\).

Case \(n\geq0\). For \(w_n(z)\) one can obtain the representation

\[ w_m(z)=e^{\omega_m(z)-p_m[\chi_m(z)]} \left\{ -\frac{1}{\pi}\iint_D \left[ \frac{f_m(t)e^{p_m(t)}}{t-\chi_m(z)} + \right.\right. \]

\[ \left.\left. + \frac{\chi_m^{2m+1}(z)\overline{f_m(t)}\,e^{\overline{p_m(t)}}}{1-\chi_m(z)\overline{t}} \right]\,dT_t + \Phi_m[\chi_m(z)] \right\}, \tag{13} \]

where \(\Phi_m(\zeta)\) is a polynomial of degree \(2n\), whose coefficients are bounded

by a number independent of \(m\); \(\omega_m(z)\) is a sequence of solutions of the problem

\[ \frac{\partial \omega_m}{\partial \bar z}+\mu_m\frac{\partial \omega_m}{\partial z}+A_m=0,\qquad \operatorname{Im}\omega_m\big|_{\Gamma}=0;\qquad \int_{\Gamma}\omega_m\,ds=0;\qquad |\mu_m|\leq \mu_0;\qquad \|A_m\|_{L}\leq 2K; \]

\(\chi_m(z)\) is a sequence of homeomorphisms of the Beltrami system

\[ \frac{\partial \chi_m}{\partial \bar z}+\mu_m\frac{\partial \chi_m}{\partial z}=0, \]

mapping the disk onto itself; \(p_m(\zeta)\) is a function analytic in \(D\);

\[ \operatorname{Im} p(\zeta)\big|_{\Gamma} = -n\arg \left.\frac{\psi_m(\zeta)}{\zeta}\right|_{\Gamma}; \]

\(\psi_m(\zeta)\) is the homeomorphism inverse to \(\chi_m(z)\), satisfying the equation

\[ \frac{\partial \psi_m}{\partial \bar\zeta} -\mu_m(\psi_m)\frac{\partial \overline{\psi_m}}{\partial \bar\zeta}=0; \]

\[ f_m(\zeta)= \frac{g_m[|\psi_m(\zeta)|]}{\|\rho\|_{L_p}}\, \frac{1}{1-|\mu_m|^2}\, \frac{1}{\partial\chi_m/\partial z}; \qquad \|f_m(\zeta)\|_{L_{p_1}}\leq \frac{K_1}{\|\rho_m\|_{L_p}}; \qquad p_1=\frac{p^2}{2(p-1)}, \]

\(2<p_1<p\); \(K_1\) depends only on \(\mu_0,K,p\). In view of the fact that \(\|\rho_m\|_{L_p}\to\infty\),

\[ \|f_m\|_{L_{p_1}}\to 0. \tag{14} \]

From the results of B. V. Boyarskii \((^3)\) it follows that the sequences \(\omega_m(z)\), \(\chi_m(z)\), \(\psi_m(z)\), \(w_m(z)\), \(p_m(\zeta)\) are bounded in norm in the space \(W_p^{(1)}(D)\), \(p>2\). From the complete continuity of the embedding operator \(W_p^{(1)}(D)\), \(p>2\), into \(C(\bar D)\) \((^4)\), it follows that these sequences are compact in \(C(\bar D)\). Let \(\omega_0(z)\), \(\chi_0(z)\), \(\psi_0(\zeta)\), \(w_0(z)\), \(p_0(\zeta)\) be limit points of our sequences in the space \(C(\bar D)\); let \(\Phi_0(\zeta)\) be a limit point in \(C(\bar D)\) of the sequence of polynomials \(\Phi_m(\zeta)\). Taking into account (13), (14), and passing to the limit in (13), we obtain the function

\[ w_0(z)=e^{\omega_0(z)-p_0[\chi_0(z)]}\Phi_0[\chi_0(z)]; \tag{15} \]

\(\chi_0(z)\) is a homeomorphism, since \(\chi_m(\psi_m(\zeta))=\zeta\). In view of uniform convergence, \(w_0(z)\) satisfies condition (5). As follows from \((^1)\), \(w_0(z)\) has on \(\Gamma\) not fewer than \(2(n+1)\) zeros. But then from (15), in view of the fact that \(\Phi_0(\zeta)\) is of full degree \(2n\), it follows that \(w_0(z)\equiv 0\).

Thus, from \(w_m(z)\) we can extract a subsequence converging uniformly to zero. Let this be the sequence \(w_m(z)\) itself. Then \(\tau_m\) converges weakly to zero, \(\|a_{im}T_n\tau_m\|_{L_p}\to 0\), \(\|\mu_{im}T_n\tau_m\|_{L_p}\to 0\). But \(\tau_m\) satisfies the equation

\[ \tau_m+\mu_{1m}S_n^0\tau_m+\mu_{2m}\overline{S_n^0}\tau_m = \frac{g_m}{\|\rho_m\|} -a_{1m}T_n\tau_m - \]

\[ {}-a_{2m}\overline{T}_n\tau_m-\mu_{1m}T_n^0\tau_m-\mu_{2m}\overline{T_n^0}\tau_m. \tag{16} \]

Choose \(p\) from the interval \(2<p<2+\varepsilon\) so that \(\mu_0\|S_n^0\|_{L_p}<1\), which can be done by virtue of the continuous dependence of \(\|S_n^0\|_{L_p}\) on \(p\) and \(\|S_n^0\|_{L_2}=1\). From equation (16) one obtains the estimate

\[ \|\tau_m\|_{L_p} \leq \frac{1}{1-\mu_0\|S_n^0\|_{L_p}} \left\| \frac{g_m}{\|\rho_m\|} -a_{1m}T_n\tau_m-a_{2m}\overline{T}_n\tau_m - \right. \]

\[ \left. {}-\mu_{1m}T_n^0\tau_m-\mu_{2m}\overline{T_n^0}\tau_m \right\|_{L_p} \to 0. \]

This contradicts the fact that \(\|\tau_m\|=1\). Consequently, for the case of nonnegative index the theorem is proved.

Case \(n<0\). The representation formula has the form

\[ w_m(z)= e^{\omega_m(z)-p_m[\chi_m(z)]} \left\{ -\frac{1}{\pi} \iint_D \left[ \frac{f_m(t)e^{p_m(t)}}{t-\chi_m(z)} + \frac{\bar t^{\,2|n|-1}\overline{f_m(t)}\,e^{\overline{p_m(t)}}}{1-\bar t\chi_m(z)} \right]\,dT_t \right\}, \]

and the remaining arguments are carried out analogously.

Theorem 2. The solutions of Problems 1 and 2 are bounded in norm in \(W_p^{(1)}(D)\), \(p>2\), by a number depending only on \(\mu_0\) and \(K\).

The proof follows immediately from Theorem 1 and the representation formulas

\[ w = T_n \rho,\qquad \frac{\partial w}{\partial \bar z} = \rho,\qquad \frac{\partial w}{\partial z} = S_n \rho . \]

In conclusion I express my gratitude to I. N. Vekua, under whose supervision this work was carried out, and to B. V. Boyarskii for valuable advice in carrying out this work.

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, Mat. Sb., 31 (73), issue 2 (1952).
\(^{3}\) B. V. Boyarskii, Mat. Sb., 43 (85), issue 4 (1957).
\(^{4}\) S. L. Sobolev, Some Applications of Functional Analysis in Mathematical Physics, 1950.