UDC 517.9

MATHEMATICS

A. DZHURAEV

ON THE THEORY OF BOUNDARY-VALUE PROBLEMS FOR SYSTEMS OF EQUATIONS OF COMPOSITE TYPE WITH MULTIPLE CHARACTERISTICS

(Presented by Academician I. N. Vekua, 5 X 1967)

Boundary-value problems for the system of first-order partial differential equations

\[ \frac{\partial w_k}{\partial \bar z} - q_k(z)\frac{\partial w_k}{\partial z} = \sum_{l=1}^{2}\left(A_{kl}(z)w_l+B_{kl}(z)\bar w_l\right), \qquad k=1,2, \tag{1} \]

in which \(w_k(z)\) are the unknown complex-valued functions, \(q_k(z)\), \(A_{kl}(z)\), \(B_{kl}(z)\) are given complex-valued functions, \(\partial/\partial \bar z=\tfrac12(\partial/\partial x+i\partial/\partial y)\), \(\partial/\partial z=\tfrac12(\partial/\partial x-i\partial/\partial y)\), in the case of its uniform ellipticity, have been studied sufficiently well \((^{1})\). Here the uniform ellipticity of system (1) means that throughout the whole domain \(|q_k(z)|\le q_k^0=\mathrm{const}<1\).

In the present paper we formulate and study a boundary-value problem for system (1), when in the entire bounded domain \(G\) under consideration in the complex plane \(z=x+iy\) the coefficients \(q_k(z)\) satisfy the conditions: \(|q_1(z)|\equiv 1\), \(|q_2(z)|\le \mathrm{const}<1\). Under these conditions system (1) is a system of composite type, i.e., it has two families of imaginary characteristics and one double family of real characteristics. Let \(\xi(x,y)=\mathrm{const}\) be an integral of the ordinary differential equation \(\operatorname{Im} q_1(z)\,dx-(1-\operatorname{Re}q_1(z))\,dy=0\). Then it is easy to verify that, as a result of the nonsingular transformation \(\xi=\xi(x,y)\), \(\eta=\eta(x,y)\), system (1) is transformed to the form

\[ \partial w_1/\partial \bar\zeta-\partial w_1/\partial \zeta = A_0(\zeta)w_1+B_0(\zeta)\bar w_1+C_0(\zeta)w_2+D_0(\zeta)\bar w_2, \]

\[ \partial w_2/\partial \bar\zeta-q(\zeta)\partial w_2/\partial \zeta = A_1(\zeta)w_1+B_1(\zeta)\bar w_1+C_1(\zeta)w_2+D_1(\zeta)\bar w_2, \tag{I} \]

where \(\partial/\partial \bar\zeta=\tfrac12(\partial/\partial \xi+i\partial/\partial \eta)\), \(\partial/\partial \zeta=\tfrac12(\partial/\partial \xi-i\partial/\partial \eta)\), \(\zeta=\xi+i\eta\); \(q(\zeta)\), \(A_j(\zeta)\), \(B_j(\zeta)\), \(C_j(\zeta)\), \(D_j(\zeta)\) are certain complex-valued functions, with \(|q(\zeta)|\le \mathrm{const}<1\). In what follows we shall consider system (I) in a certain simply connected domain \(G\) of class \(C_\nu^1\) in the plane \(\zeta=\xi+i\eta\), assuming that \(q(\zeta)\) is Hölder continuous together with its first derivatives, satisfies the inequality \(|q(\zeta)|\le \mathrm{const}<1\) in the closed domain \(\bar G=G+\Gamma\), is Hölder continuous and belongs to the class \(L_p\) \((p>2)\) in the whole \(\zeta\)-plane, while the remaining coefficients of system (I) will be assumed Hölder continuous in \(\bar G\). With respect to the domain \(G\) we shall also assume that its boundary \(\Gamma\) has tangents \(\xi=\xi_0\), \(\xi=\xi^0\) at the points \(M\) and \(N\), respectively, and that every straight line \(\xi=\xi^*\) with \(\xi_0<\xi^*<\xi^0\) intersects \(\Gamma\) in exactly two points, while the straight lines \(\xi=\xi^*\) for \(\xi^*<\xi_0\) and \(\xi^*>\xi^0\) have no common points with \(\Gamma\). Then the points \(M,N\) divide the contour \(\Gamma\) into two arcs: \(\gamma\) and \(\Gamma-\gamma\).

Problem D. Find regular solutions of system (I) in the domain \(G\), Hölder continuous in \(\bar G\) and satisfying the boundary conditions

\[ \operatorname{Re}\bigl(a_j(t)w_1(t)+b_j(t)w_2(t)\bigr)=h_j(t), \tag{D} \]

where \(t\in\Gamma\) for \(j=0\) and \(t\in\gamma\) for \(j=1,2\), and \(a_j(t)\), \(b_j(t)\) are complex-

single-valued, and \(h_j(t)\) are real functions, Hölder-continuous respectively on \(\Gamma\) for \(j=0\) and on \(\gamma\) for \(j=1,2\).

We shall study the formulated problem under the following conditions:
\[ \begin{gathered} \Delta(t)\ne 0,\quad t\in\gamma,\qquad \Delta(t)=2i\bigl(b_0(t)a_{12}(t)-b_1(t)a_{02}(t)+b_2(t)a_{01}(t)\bigr),\\ a_{ij}(t)=\operatorname{Im}\bigl(a_i(t)\overline{a_j(t)}\bigr);\qquad b_0(t)\ne 0,\quad t\in\Gamma-\gamma; \tag{N}\\ a_0(M)=a_0(N)=0,\qquad 2\operatorname{Im}\bigl(a_1(t)\overline{a_2(t)}\bigr)_{t=M,N}=-1. \tag{M} \end{gathered} \]

Along with problem \(D\), consider the homogeneous adjoint problem:

Problem \(D_0^*\). Find solutions of the system, regular in \(G\),
\[ \partial w_1^*/\partial \bar\zeta-\partial \overline{w_1^*}/\partial \zeta = \overline{A_0(\zeta)}\,w_1^* -\overline{B_0(\zeta)}\,\overline{w_1^*} +iB_1(\zeta)w_2^* +i\overline{A_1(\zeta)}\,\overline{w_2^*}, \]
\[ -\partial w_2^*/\partial\bar\zeta+\partial\bigl(q(\zeta)w_2^*\bigr)/\partial\zeta = -iD_0(\zeta)w_1^* +iC_0(\zeta)\overline{w_1^*} +C_1(\zeta)w_2^* +\overline{D_1(\zeta)}\,\overline{w_2^*}, \]
Hölder-continuous in \(G\) and satisfying the boundary conditions
\[ \operatorname{Re}\left( \frac{d(t)}{\Delta(t)}\,\xi'(s)w_1^*(t) +\frac{\theta(t)}{2i\Delta(t)}\,w_2^*(t) \right)=0,\qquad t\in\gamma, \]
\[ \left. \begin{aligned} \xi'(s)w_1^*(t) +\frac{\overline{a_0(t)}\,\overline{\theta(t)}}{2ib_0(t)}\,\overline{w_2^*(t)}&=0,\\ \operatorname{Re}\left(\frac{\theta(t)}{b_0(t)}\,w_2^*(t)\right)&=0, \end{aligned} \right\}\quad t\in\Gamma-\gamma, \tag{\(D_0^*\)} \]
where \(t=t(s)=\xi(s)+i\eta(s)\) is a parametric equation of the contour \(\Gamma\),
\[ \theta(t)=t'(s)+q(t)\overline{t'(s)},\qquad d(t)=\frac{2i}{\Delta(t)} \bigl(\overline{a_0(t)}\,b_{12}(t)-\overline{a_1(t)}\,b_{02}(t)+\overline{a_2(t)}\,b_{01}(t)\bigr), \]
\[ b_{ij}(t)=\operatorname{Im}\bigl(\overline{b_i(t)}\,b_j(t)\bigr). \]

Theorem 1. The homogeneous problem \(D_0\) \((h_j(t)\equiv0)\) and the corresponding homogeneous adjoint problem \(D_0^*\) can have only finite numbers of linearly independent solutions.

Theorem 2. In order that problem \(D\) have a solution, it is necessary and sufficient that the equalities
\[ \operatorname{Re}\int_\gamma h^{(1)}(t)\xi'(s)w_1^{(j)*}(t)\,ds +i\int_\gamma h^{(0)}(t) \left( \frac{\overline{d(t)}}{\Delta(t)}\,\xi'(s)w_1^{(j)*}(t) +\frac{\theta(t)}{2i\Delta(t)}\,w_2^{(j)*}(t) \right)\,ds \]
\[ +\int_{\Gamma-\gamma} h_0(t)\frac{\theta(t)}{2ib_0(t)}\,w_2^{(j)*}(t)\,ds=0, \qquad j=1,2,\ldots,l^*, \]
hold, where
\[ \bigl(w_1^{(1)*}(\zeta),w_2^{(1)*}(\zeta)\bigr),\ldots, \bigl(w_1^{(l^*)*}(\zeta),w_2^{(l^*)*}(\zeta)\bigr) \]
is a complete system of linearly independent solutions of the problem \(D_0^*\),
\[ h^{(0)}(t)=-2\bigl(h_0(t)a_{12}(t)-h_1(t)a_{02}(t)+h_2(t)a_{01}(t)\bigr), \]
\[ h^{(1)}(t)=\frac{3}{2\Delta(t)} \bigl(h_0(t)(a_1(t)b_2(t)-b_1(t)a_2(t)) -h_1(t)(a_0(t)b_2(t)-b_0(t)a_2(t)) +h_2(t)(a_0(t)b_1(t)-b_0(t)a_1(t))\bigr). \]

Theorem 3. The relation
\[ l-l^*=1+\frac{1}{\pi}\{\arg \overline{a^*(t)}\}_\Gamma \]
holds, where \(l\) is the number of linearly independent solutions of the homogeneous problem \(D_0\), \(l^*\) is the number of linearly independent solutions of the problem \(D_0^*\), \(a^*(t)=i\Delta(t)\) for \(t\in\gamma\) and \(a^*(t)=b_0(t)\) for \(t\in\Gamma-\gamma\), and the increment of the argument of the function \(\overline{a^*(t)}\) is computed when the contour \(\Gamma\) is traversed in the direction leaving the domain \(G\) on the left.

In Theorem 3, linear independence is understood over the field of real numbers. The proofs of the indicated theorems are based on investigations of one class of singular integro-functional equations. Let us outline the proof. First of all, note that if \(w_1^0(\zeta)\) and \(w_2^0(\zeta)\) are, respectively, solutions regular in \(G\) of the equations \(\partial w_1^0/\partial\bar\zeta-\partial w_1^0/\partial\zeta=\)

\[ = iK_0(\xi)\overline{w_1^0};\qquad \partial w_2^0/\partial \xi-q(\xi)\partial w_2^0/\partial \overline{\xi} = C_1(\xi)w_2^0+D_1(\xi)\overline{w_2^0}, \]
satisfying equalities of the form

\[ \operatorname{Re}\iint_G f(\zeta)\varphi_j(\zeta)\,dG_\zeta=0,\qquad j=1,2,\ldots,N, \]

where

\[ K_0(\xi)=-iB_0(\xi)e^{2i\operatorname{Re}p(\xi)};\qquad p(\xi)=\int_{\eta}^{\xi}A_0(\xi+i\sigma)\,d\sigma;\qquad f(\xi)=w_1^0(\xi)+ \]

\[ +\int_{\eta}^{\xi}\{R_1(\xi+i\sigma,\xi)w_2^0(\xi+i\sigma) +R_2(\xi+i\sigma,\xi)\overline{w_2^0(\xi+i\sigma)}\}\,d\sigma;\qquad R_k(\xi+i\sigma,\xi) \]

and \(\varphi_j(\zeta)\) are certain completely determined functions expressed through the coefficients of system (I), then it can be shown that all regular solutions of system (I) are representable in the form

\[ w_1(\xi)=e^{-ip(\xi)} \left\{f(\xi)+\iint_G\{k_{11}(t,\xi)f(t)+k_{12}(t,\xi)\overline{f(t)}\}\,dG_t +\sum_{j=1}^{N}C_jw_j^{(1)}(\xi)\right., \]

\[ \left. w_2(\xi)=w_2^0(\xi)+\iint_G\{k_{21}(t,\xi)f(t)+k_{22}(t,\xi)\overline{f(t)}\}\,dG_t +\sum_{j=1}^{N}C_jw_j^{(2)}(\xi), \right\} \tag{2} \]

where \(k_{ij}(t,\zeta)\), \(w_j^{(1)}(\zeta)\), \(w_j^{(2)}(\zeta)\) are completely determined functions, moreover \(w_j^{(1)}(\zeta)\), \(w_j^{(2)}(\zeta)\) are Hölder continuous in \(\overline{G}\), while \(k_{ij}(t,\zeta)\) are Hölder continuous in \(\overline{G}\), except for the point \(t=\zeta\), at which they have singularities of no more than logarithmic type. Further, it is easy to verify that the function \(w_1^0(\zeta)\) has the form
\(w_1^0(\zeta)=\Gamma_1(\zeta)\omega(\xi)+\Gamma_2(\zeta)\overline{\omega(\xi)}\), where \(\Gamma_j(\zeta)\) are known functions satisfying the condition
\(|\Gamma_1(\zeta)|^2-|\Gamma_2(\zeta)|^2\equiv 1\), and \(\omega(\xi)\) is an arbitrary complex-valued function, Hölder continuous together with its derivative on the segment \((\xi_0,\xi^0)\). Inequalities (N) make it possible to write the boundary conditions (D) in the form

\[ \operatorname{Re}\, i\Delta(t)w_2(t)=h^0(t),\qquad t\in\gamma; \tag{3} \]

\[ \operatorname{Re}\bigl(a_0(t)w_1(t)+b_0(t)w_2(t)\bigr)=h_0(t),\qquad t\in\Gamma-\gamma; \tag{4} \]

\[ w_1(t)=d(t)\overline{w_2(t)}+h^{(1)}(t),\qquad t\in\gamma. \tag{5} \]

Using now the boundary condition (5), we obtain, for determining the function \(\omega(\xi)\), an integral equation of the second kind of Fredholm type, whose right-hand side contains the function \(w_2^0(\xi)\) together with the real constants \(C_1,C_2,\ldots,C_N\). Eliminating the function \(\omega(\xi)\) from the obtained equation and representing the function \(w_2^0(\zeta)\) in the form of an integral over the contour \(\Gamma\) with unknown real density \(\mu(t)\), and using representation (2), we obtain representations of the following form for regular solutions of system (I) satisfying condition (5):

\[ w_1(\zeta)=e^{-ip(\zeta)} \{\Gamma_1(\zeta)d^0(\xi+i\sigma_1(\xi)) +\Gamma_2(\zeta)\overline{d^0(\xi+i\sigma_1(\xi))}\}\mu(\xi+i\sigma_1(\xi))+ \]

\[ +\int_{\xi_0}^{\xi^0}l_1(\xi',\xi)\mu(\xi'+i\sigma_1(\xi'))\,d\xi' +\int_{\Gamma}S^{(0)}(t,\zeta)\mu(t)\,ds +\sum_{j=0}^{n+N}C_jH_j^{(1)}(\zeta)+H^{(1)}(\zeta), \]

\[ w_2^0(\zeta)=\frac{1}{\pi i}\int_{\Gamma}Z(t,\zeta)\mu(t)\,ds +\int_{\Gamma}S^{(1)}(t,\zeta)\mu(t)\,dt+ \tag{6} \]

\[ +\int_{\xi_0}^{\xi^0}l_2(\xi',\xi)\mu(\xi'+i\sigma_1(\xi'))\,d\xi' +H^{(2)}(\zeta)+\sum_{j=0}^{n+N}C_jH_j^{(2)}(\zeta), \]

where \(\eta=\sigma_1(\xi)\) is the equation of the contour \(\gamma\);

\[ Z(t,\zeta)=\frac{\partial W/\partial t}{W(t)-W(\zeta)}; \]

\(W=W(\zeta)\) is the principal homeomorphism of the Beltrami equation (2), \(\partial W/\partial \bar{\zeta}-q(\zeta)\partial W/\partial \zeta=0\); \(S^{(k)}(t,\zeta)\) are functions satisfying the inequalities \(|S^{(k)}(t,\zeta)|\le \mathrm{const}/|t-\zeta|^\beta,\ 0\le \beta<1\); \(d^0(t)=\overline{\Gamma}_1(t)d_0(t)-\Gamma_2(t)\overline{d_0(t)}\); \(d_0(t)=e^{ip(t)}d(t)\); \(l_k(\xi',\xi)\), \(H_j^{(k)}(\xi)\) are completely determined functions independent of the right-hand sides of problem D; \(H^{(k)}(\xi)\) are completely determined functions depending on the right-hand sides of problem D; \(\mu(t)\) is an unknown real function satisfying conditions of the form

\[ \sum_{j=0}^{n+N}\tilde a_{ij}C_j=\tilde b_i,\qquad \tilde b_i=\int_{s_0}^{s}K_i^0(s',s)\mu(s')\,ds' + \]

\[ +\int_{s_1}^{s}\{K_i^{(1)}(s',s)H^{(1)}(s')+ K_i^{(2)}(s',s)H^{(2)}(s')\}\,ds', \qquad i=1,2,\ldots,n+N. \]

Substituting now the representations (6) into the boundary conditions (2) and (4), we obtain the singular integro-functional equation

\[ \operatorname{Re}a^*(t_0)\mu(t_0)+\operatorname{Re}b^*(t_0)\mu[t(\alpha(s_0))] +\frac{1}{\pi}\operatorname{Im}a^*(t_0)\int_{\Gamma}\frac{\mu(t)\,dt}{t-t_0} \]

\[ -\frac{1}{\pi}\operatorname{Im}b^*(t_0)\int_{\Gamma} \frac{\mu[t(\alpha(s))]\,dt}{t-t_0} +T(\mu)=\sum_{j=0}^{n+N}C_jH_j^*(t_0)+H^*(t_0), \qquad t_0\in\Gamma, \tag{7} \]

where \(\alpha(s)\) is a homeomorphism of the contour \(\Gamma\) onto itself, satisfying the condition \(\alpha(\alpha(s))\equiv s\); the function \(a^*(t)\) is defined in Theorem 3; \(b^*(t)=0,\ t\in\gamma\),

\[ b^*(t)=e^{-ip(t)}a_0(t)\bigl(\Gamma_1(t)d^0[t(\alpha(s))] +\Gamma_2(t)\overline{d^0[t(\alpha(s))]}\bigr); \]

\(T(\mu)\) is a completely continuous operator; \(H_j^*(t)\) are known functions independent of the right-hand sides of problem D, while \(H^*(t)\) is expressed in terms of the right-hand sides of problem D, in such a way that \(H^*(t)\equiv 0\) in the case of the homogeneous problem \(D_0\). Conditions (M) ensure the smoothness of the coefficients of equation (7). Equation (7), together with the conditions

\[ \sum_{j=0}^{n+N}\tilde a_{ij}C_j=\tilde b_i, \]

is equivalent to problem D. From the results of works \((^3,^4)\) there now follow the assertions of the theorems stated above.

Physical-Technical Institute
Academy of Sciences of the Tajik SSR

Received
2 X 1967

CITED LITERATURE

\(^1\) B. V. Boyarskii, DAN, 124, No. 1 (1958).
\(^2\) I. N. Vekua, Generalized Analytic Functions, Moscow, 1959.
\(^3\) A. Dzhuraev, DAN, 171, No. 2 (1966).
\(^4\) A. Dzhuraev, Izv. AN SSSR, Ser. Mat., 31, No. 3 (1967).