UDC 517.946.8

A GENERAL BOUNDARY VALUE PROBLEM FOR ELLIPTIC SYSTEMS OF SECOND ORDER WITH CONSTANT COEFFICIENTS

N. E. TOVMASYAN

Let \(D\) be a simply connected plane domain, and let \(\Gamma\) be its boundary. In this paper the following problem is considered: it is required to find a solution, twice continuously differentiable in the domain \(D\), of the elliptic system

\[ A u_{xx} + 2B u_{xy} + C u_{yy} = 0, \tag{1} \]

belonging to the class \(C_\alpha^n(\overline D)\) and satisfying the boundary condition

\[ a(z)u_x + b(z)u_y + c(z)u = f(z) \quad \text{on } \Gamma, \tag{2} \]

where \(u=(u_1,\ldots,u_n)\) is the unknown vector; \(f=(f_1,\ldots,f_n)\) is a given real vector on \(\Gamma\); \(A,B,C\) are real constant square matrices of order \(n\); \(a(z), b(z), c(z)\) are square matrices of order \(n\) on \(\Gamma\), \(z=x+iy\).

Let \(z=t(s)\) be a parametric equation of the contour \(\Gamma\). It is assumed that \(a(z), b(z), c(z)\), and \(f(z)\) belong to the class \(C_\alpha^n(\Gamma)\), and \(t(s)\in C_\alpha^{2n}(\Gamma)\).

We shall call problem (1), (2) Noetherian if the homogeneous problem (1), (2) has a finite number of linearly independent solutions, and for solvability of the nonhomogeneous problem (1), (2) it is necessary and sufficient that the vector-function \(f(z)\) satisfy a finite number of orthogonality conditions.

If \(u(x,y)\) and its partial derivatives up to order \(m\) satisfy a Hölder condition with exponent \(\alpha\) in the closed domain \(\overline D\), then we shall write \(u(x,y)\in C_\alpha^m(\overline D)\). If the derivatives of \(f\) with respect to \(s\) up to order \(m\) satisfy a Hölder condition with exponent \(\alpha\) on \(\Gamma\), then we shall write \(f\in C_\alpha^m(\Gamma)\) (\(s\) denotes arc length).

As is known, system (1) is called elliptic if \(\det C\ne 0\), and the characteristic equation

\[ \det(A+2B\lambda+C\lambda^2)=0 \tag{3} \]

has no real roots.

For a broad class of elliptic systems, problem (1), (2) was studied in the works [1—6]. In [7] a sufficient condition for the Noetherian property of problem (1), (2) (the E. B. Lopatinskii condition) was obtained in terms of the coefficients of system (1) and the boundary condition (2):

\[ \int_\gamma K'(z,\lambda)\Delta(\lambda)K(z,\lambda)\,d\lambda \ne 0,\quad z\in\Gamma, \tag{L} \]

where \(\Delta(\lambda)\) is the matrix inverse to the matrix \(A+2B\lambda+C\lambda^2\); \(K(z,\lambda)=\|a(z),\lambda b(z)\|\), \(K'(z,\lambda)\) is the transpose of the matrix \(K(z,\lambda)\), and \(\gamma\) is a contour in the half-plane \(\operatorname{Im}\lambda>0\), enclosing all roots of the polynomial \(\det(A+2B\lambda+C\lambda^2)\) that lie in this half-plane. If \(a(z)\equiv 0\), \(b(z)\equiv 0\), then \(K(z,\lambda)=c(z)\). We note that the problems considered in [1–6] satisfy condition \((L)\).

Below we propose a condition (more general than condition \((L)\)) whose fulfillment ensures the Noether property of problem (1), (2). In addition, a formula will be given for computing the index, i.e., the integer that is the difference between the number of solutions of the homogeneous problem corresponding to problem (1), (2) and the number of solvability conditions for the nonhomogeneous problem (1), (2).

When the matrices \(a(z)\not\equiv 0\) or \(b(z)\not\equiv 0\), the paper studies problem (1), (2) of two types: a problem satisfying condition \((L)\), and a problem that does not satisfy condition \((L)\) but nevertheless is Noetherian.

For the first type of problem (1), (2), the membership of \(a(z)\), \(b(z)\), \(c(z)\), and \(f(z)\) in the class \(C_\alpha^k(\Gamma)\) ensures that the solutions belong to the class \(C_\alpha^{k+1}(\Gamma)\) (\(k>0\)). For the second type of problems, the membership of \(a(z)\), \(b(z)\), \(c(z)\), and \(f\) in the class \(C_\alpha^k(\Gamma)\) ensures that the solutions belong to the class \(C_\alpha^k(\Gamma)\), but, generally speaking, the solutions need not belong to the class \(C_\alpha^{k+1}(\Gamma)\). In both cases it is assumed that the boundary of the domain \(D\) belongs to the class \(C_\alpha^{2k+1}\).

In [5] it was shown that if \(a(z)\), \(b(z)\), \(c(z)\), and \(f\) belong to the class \(C_\alpha^k(\Gamma)\), then a finite number of orthogonality conditions imposed on \(f\) ensures solvability of problem (1), (2) in the class \(C_\alpha^{k+1}(\Gamma)\) if and only if condition \((L)\) is fulfilled. Below it will be shown that if \(a(z)\), \(b(z)\), \(c(z)\), and \(f(z)\) belong to the class \(C_\alpha^k(\Gamma)\), and the solution is sought in the class \(C_\alpha^k(\overline D)\), then there exists a sufficiently broad class of problems (1), (2) for which condition \((L)\) is violated, but nevertheless these problems are Noetherian.

As a Noetherian problem for which condition \((L)\) is violated, one may give the following simple example. One seeks a solution, regular in the domain \(D\), of the system

\[ \frac{\partial^2 u_1}{\partial x^2}+\frac{\partial^2 u_1}{\partial y^2}=h_1(z),\qquad \frac{\partial^2 u_2}{\partial x^2}+\frac{\partial^2 u_2}{\partial y^2}=h_2(z), \]

belonging to the class \(C_\alpha^k(\overline D)\) and satisfying the boundary conditions:

\[ \left.\left(u_1+\frac{\partial u_2}{\partial N}\right)\right|_{\Gamma}=f_1,\qquad \left.\frac{\partial u_2}{\partial s}\right|_{\Gamma}=f_2, \]

where \(N\) is the inward normal to the boundary \(\Gamma\), \(h_1, h_2\in C_\alpha^{k-1}(\overline D)\), and \(f_1\) and \(f_2\in C_\alpha^k(\Gamma)\). It is obvious that, in this example, the homogeneous problem has only one linearly independent solution \(u_1=0\), \(u_2=1\), while the nonhomogeneous problem is solvable if and only if the condition

\[ \int_{\Gamma} f_2\,ds=0 \]

is fulfilled. At the same time it is not difficult to verify that condition \((L)\) is not fulfilled.

In § 1 problem (1), (2) is studied. In § 2 the problem adjoint to problem (1), (2) is constructed, and the results of § 1 are formulated in terms of the ad-

conjugate problem. In §§ 3 and 4 questions of normal solvability of problem (1), (2) are studied. In § 5 a simple method is given for solving the Dirichlet problem for system (1) in the case of a half-plane and a disk.

§ 1. STUDY OF PROBLEM (1), (2)

1. General solution of system (1). The formula for the general solution of system (1) is given in [8]. We shall present the same formula in a somewhat different form, more convenient for us. Without loss of generality, we shall assume that \(C=-E\), where \(E\) is the \(n\)-dimensional identity matrix.

System (1) is equivalent to the first-order system of equations

\[ v_y=\widetilde A v_x, \tag{4} \]

where

\[ v=\left(\frac{\partial u_1}{\partial x},\ldots,\frac{\partial u_n}{\partial x}, \frac{\partial u_1}{\partial y},\ldots,\frac{\partial u_n}{\partial y}\right),\qquad \widetilde A=\left\|\begin{array}{cc} 0 & E\\ A & 2B \end{array}\right\|. \tag{5} \]

Denote by \(\lambda_1,\ldots,\lambda_{\nu_0}\) the roots of the characteristic equation (3) with positive imaginary parts, and by \(k_1,k_2,\ldots,k_{\nu_0}\) the multiplicities of these roots.

It is known (see [10]) that there always exists a \(2n\)-dimensional constant matrix \(\widetilde B\) \((\det\widetilde B\ne0)\) such that the matrix \(\widetilde B^{-1}\widetilde A\widetilde B\) has Jordan canonical form. Since the coefficients of system (1) are real, and the system itself is elliptic, the matrix \(\widetilde B\) may be chosen so that the condition \(\overline d_k=d_{n+k}\) \((k=1,\ldots,n)\) holds, where \(d_k=(d_{1k},\ldots,d_{2n,k})\) is the \(k\)-th column of the matrix \(\widetilde B\). System (1), with respect to \(w=\widetilde B^{-1}v\), will have the form

\[ \frac{\partial w_{l_j+r}}{\partial y} = \alpha_{j,r-1}\frac{\partial w_{l_j+r-1}}{\partial x} +\lambda_j\frac{\partial w_{l_j+r}}{\partial x}, \qquad w_{n+p}=\overline w_p \tag{6} \]

\[ (r=1,\ldots,k_j,\ j=1,\ldots,\nu_0,\ l_1=0,\ l_j=k_1+\cdots+k_{j-1},\ p=1,\ldots,n), \]

where \(\alpha_{j0}=0\), \(\alpha_{jr}\ (r>0)\) are constant numbers, each of which is equal either to zero or to one, and \(\overline d_k\) and \(\overline w_r\) are complex conjugates of \(d_k\) and \(w_r\), respectively. The general solution of system (6) has the form

\[ w_{l_j+r} = \omega_{jr}^{(1)}(x+\lambda_j y) + \sum_{p=1}^{r-1}\beta_{jr}^{(p)}y^p \omega_{j,r-p}^{(p+1)}(x+\lambda_j y) \]

\[ (r=1,\ldots,k_j,\ j=1,\ldots,\nu_0), \tag{7} \]

where \(\omega_{jr}(x+\lambda_j y)\) \((r=1,\ldots,k_j)\) are arbitrary analytic functions with respect to \(x+\lambda_j y\), when \((x,y)\in D\); the upper index of the function \(\omega_{jr}(x+\lambda_j y)\) indicates the order of the derivative with respect to \(x+\lambda_j y\),

\[ p!\,\beta_{jr}^{(p)}=\alpha_{j,r-1}\alpha_{j,r-2}\cdots\alpha_{j,r-p}. \tag{8} \]

From (7) we obtain

\[ v=\operatorname{Re}\sum_{j=1}^{\nu_0}\sum_{r=1}^{k_j} \gamma_{jr}\left( \omega_{jr}^{(1)}(x+\lambda_j y) + \sum_{p=1}^{r-1}\beta_{jr}^{(p)}y^p \omega_{j,r-p}^{(p+1)}(x+\lambda_j y) \right), \tag{9} \]

\[ u=\operatorname{Re}\sum_{j=1}^{\nu_0}\sum_{r=1}^{k_j}\delta_{jr}\left(\omega_{jr}(x+\lambda_j y)+\sum_{p=1}^{r-1}\beta_{jr}^{(p)}y^p\omega_{j,r-p}^{(p)}(x+\lambda_j y)\right)+\alpha, \tag{10} \]

where \(\gamma_{jr}=d_{l_j+r}\), \(\delta_{jr}=(d_{1,l_j+r},d_{2,l_j+r},\ldots,d_{n,l_j+r})\), and \(\alpha\) is an arbitrary \(n\)-dimensional constant real vector. We note that if \(u\in C_\alpha^n(\overline D)\), then \(\omega_{jr}(x+\lambda_j y)\in C_\alpha^{\,n-r+1}(\overline D)\).

Without loss of generality we shall assume that the origin of coordinates lies inside the domain \(D\) and

\[ \omega_{jr}(0)=0\qquad (j=1,\ldots,\nu_0,\ r=1,\ldots,k_j). \tag{11} \]

Then, with the aid of formula (10), a one-to-one correspondence is established between solutions of system (1) and the quantities \(\omega_{jr}(x+\lambda_j y)\) and \(\alpha\) \((r=1,\ldots,k_j,\ j=1,\ldots,\nu_0)\).

1. Substituting the general solution (10) into the boundary condition (2) and using formula (9), we obtain, for determining the analytic functions \(\omega_{jr}(x+\lambda_j y)\), the following boundary-value problem:

\[ \operatorname{Re}\sum_{j=1}^{\nu_0}\sum_{r=1}^{k_j}\left[ \alpha_{jr}\left(\omega_{jr}^{(1)}(x+\lambda_j y)+ \sum_{p=1}^{r-1}\beta_{jr}^{(p)}y^p\omega_{j,r-p}^{(p+1)}(x+\lambda_j y)\right)+ \right. \]

\[ \left. +\,c(z)\delta_{jr}\left(\omega_{jr}(x+\lambda_j y)+ \sum_{p=1}^{r-1}\beta_{jr}^{(p)}y^p\omega_{j,r-p}^{(p)}(x+\lambda_j y)\right)\right]_{\Gamma} = \]

\[ =f(z)-c(z)\alpha, \tag{12} \]

where

\[ \alpha_{jr}=a(z)\delta_{jr}+b(z)\Theta_{jr},\qquad \Theta_{jr}=(d_{n+1,l_j+r},\ldots,d_{2n,l_j+r}). \tag{13} \]

We note that the vectors \(\delta_{jr}\), \(\Theta_{jr}\) and the numbers \(d_{jr}\) are connected by the relation

\[ \Theta_{jr}=\lambda_j\delta_{jr}+\delta_{j,r+1}d_{jr}\quad (r=1,\ldots,k_j-1),\qquad \Theta_{jk_j}=\lambda_j\delta_{jk_j}. \tag{14} \]

As a simple consequence of the integral representation of analytic functions given in [2], we obtain that any analytic function \(\omega_{jr}(x+\lambda_j y)\in C_\alpha^k(\overline D)\), \((k\geqslant 1)\), \(\omega_{jr}(0)=0\), can be represented in the form

\[ \omega_{jr}(z_j)=\int_{\Gamma}^{\circ}\left(\mu_{jr}(\tau)+Q_{jr}(\tau)\right)\ln\left(1-\frac{z_j}{\tau_j}\right)\,ds \quad (r=1,2,\ldots,k_j), \tag{15} \]

where \(z_j=x+\lambda_j y\), \(\tau=\xi+i\eta\in\Gamma\), \(\tau_j=\xi+\lambda_j\eta\), and \(Q_{jr}\) are arbitrary preassigned functions from the class \(C_\alpha^{k-1}(\Gamma)\); \(\mu_{jr}(\tau)\) are real functions from the class \(C_\alpha^{k-1}(\Gamma)\) satisfying the conditions

\[ \int_{\Gamma}\mu_{jr}(\tau)\,ds=0 \quad (r=1,\ldots,k_j,\ j=1,\ldots,\nu_0), \tag{16} \]

and the functions \(\mu_{jr}(\tau)\) are uniquely determined by \(\omega_{jr}(z_j)\) and \(Q_{jr}(\tau)\) \((r=1,\ldots,k_j,\ j=1,\ldots,\nu_0)\). Here by \(\ln\left(1-\dfrac{z_j}{\tau_j}\right)\) we shall un-

take the following branch:

\[ \ln\left(1-\frac{z_j}{\tau_j}\right) = \int_{(0,0)}^{(x,y)} \frac{dx_1+\lambda_j\,dy_1}{x_1+\lambda_j y_1-\tau_j}, \qquad (\xi,\eta)\in\Gamma,\quad (x,y)\in D, \]

where the path of integration lies in the domain \(D\).

From the integral representation (15) it follows directly that if the analytic functions
\(\omega_{jr}(z_j)\in C_a^{\,k-r+1}(\overline D)\) \((k\ge k_j)\), then they can be represented in the form (15), where this time \(\mu_{jr}(\tau)\) \((r=1,\ldots,k_j)\) are real functions of the class \(C_a^{\,k-r}(\Gamma)\) satisfying conditions (16), and the functions \(Q_{jr}(\tau)\) are expressed in terms of the functions \(\mu_{jr}(\tau)\) by means of the recurrence relations

\[ Q_{j1}(\tau)\equiv 0,\qquad Q_{jr}(\tau)= -\tau_j'\sum_{l=1}^{r-1}\beta_{jr}^{(l)}\eta^l \frac{d^l}{d\tau_j^l} \bigl[(\mu_{j,r-l}(\tau)+Q_{j,r-l}(\tau))(\tau_j')^{-1}\bigr] \tag{17} \]

\[ (r=2,\ldots,k_j,\qquad j=1,\ldots,\nu_0), \]

where the functions \(\mu_{jr}(\tau)\) are uniquely determined by
\(\omega_{j1}(z_j),\omega_{j2}(z_j),\ldots,\omega_{jk_j}(z_j)\). In (17)
\(\tau_j'\equiv\dfrac{d\tau_j}{ds}=\cos(N,\eta)-\lambda_j\cos(N,\xi)\), where \(N\) is the inward normal to the boundary \(\Gamma\) at the point \((\xi,\eta)\).

In what follows we shall always assume that in (15) the functions \(Q_{jr}(\tau)\) are expressed in terms of the functions \(\mu_{jr}(\tau)\) by means of relations (17). The following identities hold:

\[ \frac{\partial^l}{\partial z_j^l} \int_\Gamma g(\tau)\ln\left(1-\frac{z_j}{\tau_j}\right)\,ds_\tau = -\int_\Gamma \frac{\partial^{\,l-1}}{\partial\tau_j^{\,l-1}} \left(\frac{g(\tau)}{\tau_j'}\right) \frac{d\tau_j}{\tau_j-z_j}, \tag{18} \]

\[ \int_\Gamma \frac{y_0^q-\eta^q}{\tau_j-t_j} \frac{d^l}{d\tau_j^l}(g(\tau))\,d\tau_j = (-1)^l \int_\Gamma \frac{d^l}{d\tau_j^l} \left(\frac{y_0^q-\eta^q}{\tau_j-t_j}\right) g(\tau)\,d\tau_j, \tag{19} \]

where \(\tau=\xi+i\eta\in\Gamma\); \(t=x_0+iy_0\in\Gamma\); \(t_j=x_0+\lambda_j y_0\); \(z=x+iy\in D\);
\(g(\tau)\in C_a^l(\Gamma)\); \(q\) is a natural number.

From the required smoothness on \(\Gamma\) it follows that
\((y_0-\eta)(\tau_j-t_j)^{-1}\in C_a^{2n-1}(\Gamma)\) with respect to both arguments \(t\) and \(\tau\).

From (8) we obtain

\[ p\beta_{jr}^{(p)}=\beta_{jr}^{(1)}\cdot \beta_{j,r-1}^{(p-1)}, \qquad \beta_{jr}^{(1)}=\alpha_{j,r-1}. \tag{20} \]

Substituting the integral representations of the analytic functions \(\omega_{jr}(z_j)\) from (15) into (12) and using an identity of type (18) and equality (20), we obtain

\[ \lim_{z\to t}\operatorname{Re} \sum_{j=1}^{\nu_0}\sum_{r=1}^{k_j} \left\{ a_{jr}(t)\left[ \int_\Gamma \frac{\mu_{jr}(\tau)\,ds_\tau}{\tau_j-z_j} + \sum_{p=1}^{r-1} \beta_{jr}^{(p)}(y^p-\eta^p)\times \right.\right. \]

\[ \times \frac{d^p}{d\tau_j^p}\left(\frac{\mu_{jp}(\tau)+Q_{jp}(\tau)}{\tau'_j}\right)\frac{d\tau_j}{\tau_j-z_j} + c(t)\delta_{jr}\int_{\Gamma}\left(\left(-\mu_{jr}(\tau)+\right.\right. \]

\[ \left.\left. + a_{j,r-1}\cos(N,\xi)\frac{\mu_{j,r-1}(\tau)}{\tau'_j}\right) \ln\left(1-\frac{z_j}{\tau_j}\right)ds +\right. \]

\[ \left. +\sum_{p=1}^{r-1}(y^p-\eta^p)\beta_{ir}^{(p)} \frac{d^{p-1}}{d\tau_j^{p-1}} \left(\frac{\mu_{jp}(\tau)+Q_{jp}(\tau)}{\tau'_j}\right) \frac{d\tau_j}{\tau_j-z_j} \right\} = \]

\[ = -f(t)+c(t)\alpha, \tag{21} \]

\[ (t=x_0+iy_0\in\Gamma,\quad z=x+iy\in D,\quad \tau=\xi+i\eta\in\Gamma). \]

Using the Sokhotski–Plemelj formula (see [9], p. 37)

\[ \lim_{z\to t}\frac{1}{\pi i}\int_{\Gamma}\varphi(\tau)\frac{d\tau_j}{\tau_j-z_j} = \varphi(t)+\frac{1}{\pi i}\int_{\Gamma}\varphi(\tau)\frac{d\tau_j}{\tau_j-t_j}. \]

and an identity of type (19), from (21) we obtain

\[ \operatorname{Re}\sum_{j=1}^{\nu_0}\sum_{r=1}^{k_j} \left\{ a_{jr}(t)\left[(t'_j)^{-1}\mu_{jr}(t)\cdot\pi i +\int_{\Gamma}\frac{\mu_{jr}(\tau)\,ds}{\tau_j-t_j}\right] +\right. \]

\[ \left. +c(t)\delta_{jr}\int_{\Gamma} \left(-\mu_{jr}(\tau) +a_{j,r-1}\cos(N,\xi)(\tau'_j)^{-1}\mu_{j,r-1}(\tau)\right) \ln\left(1-\frac{t_j}{z_j}\right)ds +\right. \]

\[ \left. +\int_{\Gamma}K_{jr}(t,\tau)\mu_{jr}(\tau)\,d\tau \right\} = -f(t)+c(t)\alpha, \tag{22} \]

where \(t'_j=\dfrac{dt_j}{ds}\), \(K_{jr}(t,\tau)\in C_\alpha^n(\Gamma)\) with respect to \(t\) and \(\tau\), and all singular integrals are understood in the sense of the Cauchy principal value. It is easy to verify that

\[ \left(\ln\left(1-\frac{t_j}{\tau_j}\right)-\ln\left(1-\frac{t}{\tau}\right)\right)\in C_\alpha^n(\Gamma), \qquad \frac{t'}{\tau-t}-\frac{\tau'_j}{\tau_j-t_j}\in C_\alpha^n(\Gamma) \tag{23} \]

with respect to the variables \(t\) and \(\tau\).

Introduce the following notation:

\[ b_{l_j+r}(\tau)=\pi i\,(a(\tau)\delta_{jr}+b(\tau)\Theta_{jr})(\tau'_j)^{-1}, \qquad \widetilde{\mu}_{l_j+r}=\mu_{jr}, \tag{24} \]

\[ \sigma_{l_j+r}(\tau) = \left(\delta_{jr}\cos(N,\eta)-\Theta_{jr}\cos(N,\xi)\right) \left(\cos(N,\eta)-\lambda_j\cos(N,\xi)\right)^{-1}\cdot\pi i. \tag{25} \]

Keeping in mind (23), (14), and the notation (24), (25), we rewrite the system of integral equations (22) in the form

\[ \operatorname{Re}\sum_{r=1}^{n}\left[ b_r(t)\widetilde{\mu}_r(t)+\frac{1}{\pi i}\int_{\Gamma} \frac{b_r(\tau)\widetilde{\mu}_r(\tau)\,d\tau}{\tau-t} -\frac{c(t)\sigma_r(t)}{\pi i}\int_{\Gamma}\widetilde{\mu}_r(\tau)\ln\left(1-\frac{t}{\tau}\right)\,ds +\frac{1}{\pi i}\int_{\Gamma}c(t)(\sigma_r(t)-\sigma_r(\tau))\widetilde{\mu}_r(\tau)\ln\left(1-\frac{t}{\tau}\right)\,ds +\int_{\Gamma}p_r(t,\tau)\widetilde{\mu}_r(\tau)\,d\tau \right] =-f(t)+c(t)a, \tag{26} \]

where the \(p_r(t,\tau)\) are completely determined vectors belonging to the class \(C_\alpha^n(\Gamma)\) with respect to \(t\) and \(\tau\).

We impose on the vectors \(b_1(t),\ldots,b_n(t)\) and \(c(t)\sigma_1(t),\ldots,c(t)\sigma_n(t)\) the following conditions:

Condition 1. From the vectors \(b_1(t),\ldots,b_n(t)\) one can choose vectors
\(b_{j_1}(t),\ldots,b_{j_m}(t)\) \((0\le m\le n)\), which are linearly independent at every point \(t\in\Gamma\), while the remaining vectors \(b_r(t)\) \((r\ne j_p,\ p=1,2,\ldots,m)\) are linearly dependent at every point \(t\in\Gamma\) on the vectors \(b_{j_1}(t),\ldots,b_{j_m}(t)\).

In particular, condition 1 is always satisfied if the matrices \(a(z)\) and \(b(z)\) of the boundary condition (2) are constant, or if the boundary condition (2) is written in the form

\[ \frac{\partial u}{\partial N}+c(z)u=f(z)\quad \text{on } \Gamma . \tag{27} \]

Denote by
\[ k_1(t)\equiv(k_{11}(t),\ldots,k_{1n}(t)),\ldots, k_{n-m}(t)\equiv(k_{n-m,1}(t),\ldots,k_{n-m,n}(t)) \]
linearly independent solutions of the system of algebraic equations

\[ b_1(t)x_1+b_2(t)x_2+\cdots+b_n(t)x_n=0. \tag{28} \]

Denote by \(\sigma(t)\) the matrix whose columns are the vectors
\(\sigma_1(t),\sigma_2(t),\ldots,\sigma_n(t)\), and by \(\widetilde{k}(t)\) the matrix whose columns are the vectors
\(k_1(t),\ldots,k_{n-m}(t)\). Denote by \(G(t)\) the square matrix of order \(n\) whose columns are the vectors
\(b_{j_1}(t), b_{j_2}(t),\ldots,b_{j_m}(t)\) and the columns of the matrix \(c(t)\sigma(t)\widetilde{k}(t)\).

If \(m=n\), then the columns of the matrix \(G\) are the vectors
\(b_1(t), b_2(t),\ldots,b_n(t)\). If, however, \(m=0\), then
\(G=c(t)\sigma(t)\).

Condition 2. \(\det G(t)\ne 0,\ t\in\Gamma\).

We note that condition 2 does not depend on the choice of the linearly independent vectors
\(b_{j_1}(t), b_{j_2}(t),\ldots,b_{j_m}(t)\) nor on the choice of the linearly independent solutions of the algebraic system (28).

We rewrite the system (28) in the form

\[ \sum_{k=1}^{m} b_{j_k}x_{j_k} +\sum_{k=1}^{\,n-m} b_{i_k}x_{i_k}=0. \tag{29} \]

Since the vectors \(b_{j_1}(t),\ldots,b_{j_m}(t)\) are linearly independent at every point
\(t\in\Gamma\), we can always take as \(k_1(t),\ldots,k_{n-m}(t)\) the solution of the system (29) for which the vector
\((x_{i_1},x_{i_2},\ldots,x_{i_{n-m}})\) is respectively equal to
\((1,0,\ldots,0), (0,1,0,\ldots,0),\ldots,(0,\ldots,0,1)\). In what follows we shall choose the linearly independent solutions of the system (28) in this way. It is clear that

under such a choice the vectors \(k_1(t), \ldots, k_n(t)\) belong to the class \(C_\alpha^n(\Gamma)\).

We note that condition \((L)\) is equivalent to conditions 1 and 2 with the additional condition \(m=n\) or \(m=0\).

Theorem 1. If conditions 1 and 2 are fulfilled, then problem (1), (2) is Noetherian, and the index of problem (1), (2) is equal to

\[ -\frac{1}{\pi}\left[\arg \det G(t)\right]_{\Gamma}, \tag{30} \]

where the symbol \([\ \ ]_{\Gamma}\) denotes the increment of the expression standing in square brackets when the contour \(\Gamma\) is traversed once in the positive direction.

Proof. For simplicity we shall assume that the vectors \(b_1(t), b_2(t), \ldots, b_m(t)\) are linearly independent at every point \(t \in \Gamma\), while the vectors \(b_{m+1}(t), b_{m+2}(t), \ldots, b_n(t)\) at every point \(t \in \Gamma\) are linearly dependent on the vectors \(b_1(t), \ldots, b_m(t)\), i.e.

\[ b_r(t)=\sum_{j=1}^{m} k_{rj}(t)b_j(t), \qquad r=m+1, m+2, \ldots, n . \tag{31} \]

Using (31), equation (26) can be rewritten in the form

\[ \begin{aligned} \operatorname{Re}\Bigg\{& \sum_{j=1}^{m} b_j(t)\Bigg[ \tilde{\mu}_j(t) +\sum_{r=m+1}^{n} k_{rj}(t)\tilde{\mu}_r(t) +\frac{1}{\pi i}\int_{\Gamma} \left(\tilde{\mu}_j(\tau)+ \sum_{r=m+1}^{n} k_{rj}(\tau)\tilde{\mu}_r(\tau)\right) \frac{d\tau}{\tau-t} \Bigg] \\ &-\frac{1}{\pi i}\sum_{j=1}^{m} c(t)\sigma_j(t)(t_s')^{-1} \int_{\Gamma} \left(\tilde{\mu}_j(\tau)+ \sum_{r=m+1}^{n} k_{rj}(\tau)\tilde{\mu}_r(\tau)\right) \ln\left(1-\frac{t}{\tau}\right)\,d\tau \\ &-\frac{1}{\pi i}\sum_{j=m+1}^{n} c(t)\tilde{\sigma}_j(t) \int_{\Gamma} \tilde{\mu}_j(\tau)\ln\left(1-\frac{t}{\tau}\right)\,ds_\tau +\sum_{r=1}^{n}\int_{\Gamma}\tilde{P}_r(t,\tau)\tilde{\mu}_r(\tau)\,d\tau \\ &+\sum_{r=1}^{n}\frac{1}{\pi i}\int_{\Gamma} c(t)\bigl(\tilde{\sigma}_r(t)-\sigma_r(\tau)\bigr)\tilde{\mu}_r(\tau) \ln\left(1-\frac{\tau}{t}\right)\,ds_\tau \Bigg\} \\ &= -f(t)+c(t)a, \end{aligned} \tag{32} \]

where

\[ \tilde{\sigma}_r(t)=\sigma_r(t)-\sum_{j=1}^{m} k_{rj}(t)\sigma_j(t) \qquad (r=m+1,\ldots,n), \tag{33} \]

and \(\tilde{P}_r(t,\tau)\) are well-defined vectors of the class \(C_\alpha^n(\Gamma)\).

Let \(v_1(t), \ldots, v_n(t)\) be real functions on \(\Gamma\) which are related to \(\tilde{\mu}_1(t), \ldots, \tilde{\mu}_n(t)\) by the equalities

\[ \gamma_j(t)-\tilde{\mu}_j(t)+\frac{1}{\pi}\int_\Gamma \bigl(\gamma_j(\tau)-\tilde{\mu}_j(\tau)\bigr)\,d\nu(\tau,t)= \]

\[ =\operatorname{Re}\sum_{r=m+1}^{n}\left( k_{jr}(t)\tilde{\mu}_r(t)+\frac{1}{\pi i}\int_\Gamma \frac{k_{jr}(\tau)\tilde{\mu}_r(\tau)}{\tau-t}\,d\tau \right)\qquad (j=1,\ldots,m), \tag{34} \]

\[ \frac{dv_j(t)}{ds}=\tilde{\mu}_j(t)\qquad (j=m+1,\ m+2,\ldots,n), \tag{35} \]

and satisfy the conditions

\[ \int_\Gamma v_j(t)\,ds=0\qquad (j=m+1,\ m+2,\ldots,n), \tag{36} \]

where \(\nu(\tau,t)=\arg(\tau-t)\), \(t\in\Gamma\); \(\tau\) is the point of integration.

It is known (see [9], p. 252) that equation (34), with respect to \(\gamma_j(t)-\tilde{\mu}_j(t)\), always has a solution, and moreover a unique one. Consequently, in order to express \(v_1,\ldots,v_n\) in terms of \(\tilde{\mu}_1,\ldots,\tilde{\mu}_n\) and conversely, it suffices to construct the resolvent of the Fredholm equation

\[ \mu(t)+\frac{1}{\pi}\int_\Gamma \mu(\tau)\,d\nu(\tau,t)=f(t). \]

Since the functions \(\tilde{\mu}_1,\ldots,\tilde{\mu}_n\) satisfy conditions (16) and \(\tilde{\mu}_r(t)\in C_\alpha(\Gamma)\), it follows that \(v_r(t)\in C_\alpha(\Gamma)\), and they must satisfy conditions of the form

\[ \int_\Gamma v_j(t)\,ds_t+ \sum_{k=m+1}^{n}\int_\Gamma \alpha_{jk}(t)v_k(t)\,ds_t=0 \qquad (j=1,\ldots,m), \tag{37} \]

where \(\alpha_{jk}(t)\) are completely determined real functions.

To obtain conditions (37), one must, by means of (34)—(36), express the functions \(\tilde{\mu}_1(t),\ldots,\tilde{\mu}_n(t)\) in terms of \(v_1(t),\ldots,v_n(t)\) and subject them to conditions (16).

Equations (34), (35) establish a mutually one-to-one correspondence between the real functions \(\tilde{\mu}_1(t),\ldots,\tilde{\mu}_n(t)\), satisfying conditions (16), and the real functions \(v_1(t),\ldots,v_n(t)\), satisfying conditions (36), (37).

From (34) we obtain

\[ \frac{1}{\pi i}\int_\Gamma \frac{v_j(\tau)\,d\tau}{\tau-z} = \frac{1}{\pi i}\int_\Gamma \left(\tilde{\mu}_j(\tau)+\sum_{r=m+1}^{n}k_{jr}(\tau)\tilde{\mu}_r(\tau)\right) \frac{d\tau}{\tau-z} +ic_j,\quad z\in D \tag{38} \]

\[ (j=1,\ldots,m), \]

where

\[ c_j=\operatorname{Im}\frac{1}{\pi i}\int_\Gamma \left(\gamma_j(\tau)-\tilde{\mu}_j(\tau)- \sum_{r=m+1}^{n}k_{jr}(\tau)\tilde{\mu}_r(\tau)\right) \frac{d\tau}{\tau}. \tag{39} \]

Equality (38) follows from the fact that both sides of it are

analytic functions in the domain \(D\), and, according to (34), the real parts of these analytic functions are equal on \(\Gamma\).

Integrating (38) with respect to \(z\), we obtain

\[ \frac{1}{\pi i}\int_{\Gamma} v_j(\tau)\ln\left(1-\frac{z}{\tau}\right)d\tau = \frac{1}{\pi i}\int_{\Gamma}\left(\tilde{\mu}_j(\tau)+ \right. \]
\[ \left. +\sum_{r=m+1}^{n} k_{jr}(\tau)\tilde{\mu}_r(\tau)\right) \ln\left(1-\frac{z}{\tau}\right)d\tau +ic_j z \tag{40} \]

\[ (j=1,2,\ldots,m). \]

Passing to the limit in (38) and (40), when \(z\to t\in\Gamma\), we obtain

\[ v_j(t)+\frac{1}{\pi i}\int_{\Gamma}\frac{v_j(\tau)d\tau}{\tau-t} = \tilde{\mu}_j(t)+ \sum_{r=m+1}^{n} k_{jr}(t)\tilde{\mu}_r(t)+ic_j+ \]
\[ +\frac{1}{\pi i}\int_{\Gamma} \left(\tilde{\mu}_j(\tau)+ \sum_{r=m+1}^{n} k_{jr}(\tau)\tilde{\mu}_r(\tau)\right) \frac{d\tau}{\tau-t} \qquad (j=1,\ldots,m), \tag{41} \]

\[ \frac{1}{\pi i}\int_{\Gamma} v_j(\tau)\ln\left(1-\frac{t}{\tau}\right)d\tau = \frac{1}{\pi i}\int_{\Gamma} \left(\tilde{\mu}_j(\tau)+ \right. \]
\[ \left. +\sum_{r=m+1}^{n} k_{jr}(\tau)\tilde{\mu}_r(\tau)\right) \ln\left(1-\frac{t}{\tau}\right)d\tau +ic_j t \qquad (j=1,\ldots,m). \tag{42} \]

The identity holds

\[ \frac{1}{\pi i}\int_{\Gamma}\ln\left(1-\frac{t}{\tau}\right)dv_j(\tau) = -v_j(t)-\frac{1}{\pi i}\int_{\Gamma}\frac{v_j(\tau)d\tau}{\tau-t} + \]
\[ +\frac{1}{\pi i}\int_{\Gamma} v_j(\tau)\tau^{-1}d\tau,\qquad t\in\Gamma. \tag{43} \]

In (32), with the aid of (34), (35), replacing the unknown functions \(\tilde{\mu}_1,\ldots,\tilde{\mu}_n\) by \(v_1(t),\ldots,v_n(t)\) and using (41)—(43), we obtain

\[ \operatorname{Re}\left\{ \sum_{j=1}^{m} b_j(t)\left(v_j(t)+ \frac{1}{\pi i}\int_{\Gamma}\frac{v_j(\tau)d\tau}{\tau-t}\right) + \sum_{r=m+1}^{n} c(t)\tilde{\sigma}_r(t)\left(v_r(t)+ \right. \]
\[ \left. +\frac{1}{\pi i}\int_{\Gamma}\frac{v_r(\tau)d\tau}{\tau-t}\right) + \sum_{r=1}^{n}\tilde{Q}_r(t,\tau)v_r(\tau)d\tau \right\} = -f(t)+c(t)\alpha, \tag{44} \]

where \(\tilde{Q}_r(t,\tau)\) are completely determined vectors for which the estimate holds

\[ |\tilde{Q}_r(t,\tau)|<k|\ln(t-\tau)|\qquad (k=\mathrm{const}). \tag{45} \]

The system (38) can be rewritten in the form

\[ \sum_{j=1}^{m}\left[\operatorname{Re}(b_j(t))v_j(t)+ \frac{\operatorname{Im}(b_j(t))}{\pi}\int_{\Gamma}\frac{v_j(\tau)\,d\tau}{\tau-t}\right] + \sum_{j=m+1}^{n}\left[c(t)\operatorname{Re}(\widetilde{\sigma}_j(t))v_j(t)+ \right. \]
\[ \left. + \frac{c(t)\operatorname{Im}(\widetilde{\sigma}_j(t))}{\pi} \int_{\Gamma}\frac{v_j(\tau)\,d\tau}{\tau-t}\right] + \sum_{j=1}^{n}\widetilde{Q}_j(t,\tau)v_j(\tau)\,d\tau = -f(t)+c(t)\alpha, \tag{46} \]

where the vector functions \(\widetilde{Q}_j(t,\tau)\) satisfy the estimate (45).

It follows from Condition 2 that the system of integral equations (46) is a system of normal type. Therefore, from the smoothness conditions imposed on the boundary \(\Gamma\), on the coefficients of the boundary condition (2), and on the given function \(f(s)\), it follows that all solutions of this system belong to the class \(C_\alpha^h(\Gamma)\), and the function \(u(x,y)\), which is expressed through the solutions \(v_1(t),\ldots,v_n(t)\) of the system (46) by means of (10), (15), (17), (34), (35), belongs to the class \(C_\alpha^n(\overline{D})\). Consequently, the problem (1), (2) is equivalent to the system of integral equations (46) with the additional conditions (36), (37). Hence, by virtue of the known theorems of the theory of singular integral equations, it follows that Conditions 1 and 2 ensure the Noetherian character of the problem (1), (2).

Let \(k\) be the number of linearly independent solutions of the homogeneous equation (46) (i.e., \(c(t)\alpha-f(t)\equiv 0\)), and let \(k'\) be the number of conditions of the form

\[ \int_{\Gamma}(-f(t)+c(t)\alpha)\psi_j(t)\,ds=0 \qquad (j=1,\ldots,k'), \]

which are necessary and sufficient for the solvability of the nonhomogeneous system of equations (46). Here \(\psi_j(t)\) \((j=1,\ldots,k')\) are linearly independent real vectors.

If no additional conditions (36), (37) are imposed on the solutions \(v_1(t),\ldots,v_n(t)\) of the system of equations (46), then from the third fundamental theorem of the theory of singular integral equations (see [9], p. 510) we obtain

\[ k-k'=\chi, \tag{47} \]

where

\[ \chi=-\frac{1}{\pi}\left[\arg\det G(t)\right]_{\Gamma}, \tag{48} \]

and \(G(t)\) is the matrix whose columns are the vectors \(b_1(t),\ldots,b_m(t)\) and \(c(t)\widetilde{\sigma}_{m+1}(t),\ldots,c(t)\widetilde{\sigma}_n(t)\).

If the real functions \(v_1(t),\ldots,v_n(t)\) are subject to the conditions (36), (37), then, using the fact indicated above, it is easy to prove that

\[ k-k'=\chi-n. \tag{49} \]

It is clear that, by means of (15)—(17) and (34), (35), a one-to-one correspondence is established between the real functions \(v_1,\ldots,v_n\), satisfying the conditions (36), (37), and the analytic functions \(\omega_{jr}(x+\lambda_jy)\) \((r=1,\ldots,k_j,\ j=1,\ldots,\nu_0)\), satisfying condition (11). On the other hand, formula (10) contains an \(n\)-dimensional real constant vector \(\alpha\), and by means of this formula

a mutually one-to-one correspondence is established between solutions of system (1) and \(\omega_{jr}(x+\lambda_j y)\) \((\omega_{jr}(0)=0), \alpha\) \((r=1,\ldots,k_j,\ j=1,\ldots,\nu_0)\). Hence, from (49) it follows that the index of problem (1), (2) is equal to \(\varkappa\). The theorem is proved.

Let us note that problem (1), (2) is equivalent to the system of integral equations (46) also in the case when the real functions \(v_1(t),\ldots,v_n(t)\) are not subject to conditions (36), (37). Conditions (36), (37) are used only for computing the index of the given problem.

Following A. V. Bitsadze (see [8]), we shall call system (1) weakly coupled if the vectors \(\delta_{jr}\) \((r=1,\ldots,k_j,\ j=1,\ldots,\nu_0)\) entering into the general solution (10) are linearly independent. If the system is not weakly coupled, then we shall call it strongly coupled.

To clarify how much broader the conditions 1 and 2 given by us are than condition \((L)\), consider problem (1), (2) when system (1) is strongly coupled and the boundary condition (2) has the form (27). It can be shown that for any strongly coupled system (1) and for any boundary condition of the form (27) this problem does not satisfy condition \((L)\).

Let us determine when this problem satisfies conditions 1 and 2. Introduce the following new notation. Denote \(\delta_{jr}\) by \(\tilde{\delta}_{l_j+r}\) \((r=1,\ldots,k_j,\ j=1,\ldots,\nu_0,\ l_1=0,\ l_j=k_1+\cdots+k_{j-1})\), and \(\alpha_{jr}\) by \(\tilde{\alpha}_{l_j+i}\) \((i=0,1,\ldots,k_j-1,\ j=1,\ldots,\nu_0)\). Here \(\alpha_{jr}\) are the numbers entering into (6). Let \(m\) be the number of linearly independent vectors \(\tilde{\delta}_1,\ldots,\tilde{\delta}_n\), and let \(\tilde{\delta}_{n_1},\ldots,\tilde{\delta}_{n_m}\) be linearly independent \((n_1<n_2<\cdots<n_m\le n)\). Condition 1 for this problem is always fulfilled, and condition 2 coincides with the condition

\[ \det G_1(z)\ne 0,\qquad z\in \Gamma, \tag{50} \]

where \(G_1(z)\) is the matrix whose columns are the vectors \(\tilde{\delta}_{n_1},\tilde{\delta}_{n_2},\ldots,\tilde{\delta}_{n_m}\) and the columns of the matrix \(c(z)D^{-1}(z)c_1\). Here \(c(z)\) is the matrix entering into (27), \(c_1\) is a constant \((n-m)\times n\)-matrix whose columns are linearly independent solutions \(x=(x_1,\ldots,x_n)\) of the algebraic system

\[ \tilde{\delta}_1 x_1+\cdots+\tilde{\delta}_n x_n=0, \]

\(D^{-1}(z)\) is the \(n\times n\)-matrix inverse to the matrix \(D(z)=\|\tilde d_{jr}\|\), where

\[ \tilde d_{jj}= \bigl(\cos(N,x)+\tilde\lambda_j\cos(N,y)\bigr) \bigl(\cos(N,y)-\tilde\lambda_j\cos(N,x)\bigr)^{-1} \qquad (j=1,2,\ldots,n), \]

\[ \tilde d_{j,j-1}= \tilde\alpha_{j-1}\cos(N,y) \bigl(\cos(N,y)-\tilde\lambda_j\cos(N,x)\bigr)^{-1} \qquad (j=2,3,\ldots,n), \]

\[ \tilde\lambda_{l_r+1}=\tilde\lambda_{l_r+2}=\cdots=\tilde\lambda_{l_r+1}=\lambda_r \qquad (r=1,\ldots,\nu_0), \]

and the remaining elements of the matrix \(D(z)\) are equal to zero.

When condition (50) is fulfilled, the problem considered is Noetherian and its index is equal to

\[ -\frac{1}{\pi}\left[\arg\det G_1(z)\right]_{\Gamma}. \]

From Theorem 1 it follows:

Corollary 1. If system (1) is weakly coupled, then the Dirichlet problem and the Neumann problem for this system are Fredholm.

Remark 1. If, in the statement of problem (1), (2), one requires that
\(f\in C_{\alpha}^{2k_0}(\Gamma)\), \(t(s)\in C_{\alpha}^{2k_0}\), \(a(z), b(z), c(z), u\in C_{\alpha}^{k_0}(\overline D)\), where \(k_0\) is the maximum

orders of the highest derivatives of the functions \(\omega_{jr}(z_j)\) entering into the general representation (10), the results of the present section remain unchanged.

The methods and results of the present section carry over, without essential changes, to the case of a multiply connected domain.

§ 2. THE ADJOINT BOUNDARY VALUE PROBLEM

In this section we shall construct the boundary value problem adjoint to the following problem.

Problem A. Find, in the domain \(D\), a twice continuously differentiable solution of the elliptic system

\[ Z(u)\equiv \sum_{i+j\leq 2} A_{ij}\frac{\partial^{i+j}u}{\partial x^i\partial y^j}=h(x,y), \tag{51} \]

belonging to the class \(C_\alpha^k(\overline D)\) \((k\geq 1)\) and satisfying the boundary condition

\[ a(z)\frac{\partial u}{\partial x}+b(z)\frac{\partial u}{\partial y}+c(z)u=0 \qquad \text{on } \Gamma, \tag{52} \]

where \(u=(u_1,\ldots,u_n)\) is the unknown vector; \(h=(h_1,\ldots,h_n)\) is a prescribed real vector; \(A_{ij}\) are square matrices of order \(n\), whose elements are real functions of \((x,y)\); \(a(z)\), \(b(z)\), and \(c(z)\) are real matrices of order \(n\), prescribed on \(\Gamma\).

It is assumed that the matrices \(A_{ij}\) belong to the class \(C_\alpha^{i+j+k}(\overline D)\); \(h\in C_\alpha^{k-1}(\overline D)\), while \(a(z)\), \(b(z)\), and \(c(z)\) belong to the class \(C_\alpha^k(\Gamma)\).

After constructing the adjoint problem we shall formulate the theorem obtained in § 1 in terms of this problem.

Consider the equation adjoint to (1)

\[ Z^*(v)\equiv \sum_{i+j\leq 2}(-1)^{i+j}\frac{\partial^{i+j}vA_{ij}}{\partial x^i\partial y^j}=0. \tag{53} \]

Definition 1. The boundary condition

\[ M^*v=0 \qquad \text{on } \Gamma \tag{54} \]

will be called adjoint to the boundary condition (52) with respect to the operator \(Z(u)\), if the equality

\[ \iint_D (v,Z(u))\,dx\,dy = \iint_D (Z^*(v),u)\,dx\,dy, \qquad \left(u,v\in C^2(\overline D)\right) \tag{55} \]

holds for any vector-function \(u(x,y)\) satisfying condition (52) on \(\Gamma\), if and only if on the boundary \(\Gamma\) the vector-function \(v\) satisfies condition (54) (see [11]). Here \((v,Z(u))\) denotes the scalar product of the vector \(v\) with the vector \(Z(u)\).

We shall call problem (53), (54) adjoint to problem (52)—(51). From solutions of the adjoint problem (53), (54) the same smoothness is required as for problem (51), (52).

For the case when system (51) consists of a single equation and the coefficients \(a\) and \(b\) of the boundary condition (52) satisfy the condition

\[ a(z)\cos (N,x)+b(z)\cos (N,y)\ne 0,\qquad z\in\Gamma, \tag{56} \]

where \(N\) is the inward normal to the contour \(\Gamma\) at the point \(z\); the problem adjoint to problem (51), (52) was constructed in [11].

For the case when system (51) consists of \(n\) equations and the boundary condition (52) satisfies the condition

\[ \det \bigl(a(z)\cos (N,x)+b(z)\cos (N,y)\bigr)\ne 0,\qquad z\in\Gamma, \]

the adjoint problem can be constructed in the same way as in [11].

Below we indicate a new method for constructing the problem adjoint to problem (51), (52), if the condition

\[ \operatorname{rang}\|a(z),\, b(z)\|=n \tag{57} \]

is satisfied for every point \(z\in\Gamma\).

This condition is equivalent to saying that the rank of the matrix

\[ \|a(z)\cos (N,x)+b(z)\cos (N,y),\quad a(z)\cos (N,y)-b(z)\cos (N,x)\| \]

is equal to \(n\) for every point \(z\in\Gamma\).

In particular, condition (57) is always fulfilled for those problems which satisfy condition \((L)\).

Assume that condition (57) is fulfilled. Let

\[ \left. \left( a_1(z)\frac{\partial u}{\partial x} + b_1(z)\frac{\partial u}{\partial y} + c_1(z)u \right) \right|_{\Gamma} = g(z), \tag{58} \]

where \(a_1(z)\), \(b_1(z)\), and \(c_1(z)\) are \(n\)-dimensional square matrices whose elements belong to the class \(C_\alpha^k(\Gamma)\) and satisfy the condition

\[ \det \begin{Vmatrix} a,\ b\\ a_1,\ b_1 \end{Vmatrix} \ne 0,\qquad z\in\Gamma. \tag{59} \]

In view of condition (57), one can always find matrices \(a_1(z)\) and \(b_1(z)\) satisfying condition (59).

Let us determine what values the vector-function \(g\) can assume when we take all vector-functions \(u(x,y)\) belonging to the class \(C^2(\bar D)\) and satisfying condition (52).

Solving (52) and (58) with respect to \(\dfrac{\partial u}{\partial x}\) and \(\dfrac{\partial u}{\partial y}\), we obtain

\[ \frac{\partial u}{\partial x} = \tilde a_1 u+\tilde b_1 g, \qquad (x,y)\in\Gamma, \tag{60} \]

\[ \frac{\partial u}{\partial y} = \tilde a_2 u+\tilde b_2 g, \qquad (x,y)\in\Gamma. \tag{61} \]

From (60) and (61) we have

\[ \frac{\partial u}{\partial s} = a_3u+b_3g, \qquad (x,y)\in\Gamma, \tag{62} \]

where \(a_3=\tilde a_1\cos (N,y)-\tilde a_2\cos (N,x)\), \(b_3=\tilde b_1\cos (N,y)-\tilde b_2\cos (N,x)\).

For simplicity we shall assume that the domain \(D\) is simply connected. By \(s\) (sometimes by \(\sigma\)) we denote the length of the arc, measured from

some fixed point \(z_0 \in \Gamma\) in the positive direction. The vector \(u(x,y)\) on the boundary of the domain \(D\) may be regarded as a vector-function of \(s\) or \(\sigma\). Then (62) may be written in the form

\[ \frac{du(s)}{ds}=a_3(s)u(s)+b_3(s)g(s)\qquad (0\leq s\leq l), \tag{63} \]

where \(l\) is the length of the contour \(\Gamma\).

The solution \(u(s)\) of equation (63) has the form

\[ u(s)=\int_0^s K(s,\sigma)g(\sigma)\,d\sigma+\sum_{k=1}^n c_k u_k(s), \tag{64} \]

where \(K(s,\sigma)\) is an \(n\)-dimensional matrix whose entries, with respect to \(s\) and \(\sigma\), belong to the classes \(C_\alpha^2(\Gamma)\) and \(C_\alpha^1(\Gamma)\); \(u_1(s),\ldots,u_n(s)\) are linearly independent solutions of the homogeneous equation (63), and \(c_1,\ldots,c_n\) are arbitrary constants.

Since \(u(0)=u(l)\), from (64), in order to determine \(c_1,\ldots,c_n\), we obtain a system of algebraic equations

\[ \sum_{k=0}^n c_k\bigl(u_k(0)-u_k(l)\bigr) =\int_\Gamma K(l,\sigma)g(\sigma)\,d\sigma . \tag{65} \]

If by \(r\) we denote the number of linearly independent vectors
\(u_1(0)-u_1(l),\ u_2(0)-u_2(l),\ldots,\ u_n(0)-u_n(l)\), then, as is known, the homogeneous system (65) has \(n-r\) linearly independent solutions, and for the solvability of the nonhomogeneous system (65) it is necessary and sufficient that

\[ \int_\Gamma c_i^* K(l,\sigma)g(\sigma)\,d\sigma=0 \qquad (i=1,2,\ldots,n-r), \tag{66} \]

where \(c_1^*,\ldots,c_{n-r}^*\) are linearly independent \(n\)-dimensional vectors that are solutions of the adjoint homogeneous system.

If condition (66) is satisfied, then the general solution \(u(s)\) of system (63), satisfying the condition \(u(0)=u(l)\), will have the form

\[ u(s)=\int_0^s K(s,\sigma)g(\sigma)\,d\sigma +\int_\Gamma K_1(s,\sigma)g(\sigma)\,d\sigma +\sum_{k=1}^{n-r} c_k \widetilde{u}_k(s), \tag{67} \]

where \(\widetilde{u}_k(s)\) \((k=1,\ldots,n-r)\) are completely determined linearly independent solutions of the homogeneous system (63), and \(c_1,c_2,\ldots,c_{n-r}\) are arbitrary constants. From (60)—(67) it follows that \(g(\sigma)\) may be any \(n\)-dimensional vector-function of class \(C^1(\Gamma)\) orthogonal to the vectors \(c_i^*K(l,\sigma)\) \((i=1,2,\ldots,n-r)\), and the functions \(u(s)\), \(\left.\dfrac{\partial u}{\partial x}\right|_\Gamma\), \(\left.\dfrac{\partial u}{\partial y}\right|_\Gamma\) are expressed in terms of \(g(s)\) by means of formulas (60), (61), and (67).

Let \(v(x,y)\) be an \(n\)-dimensional vector-function of class \(C^2(\overline D)\), for which equality (55) holds for any \(n\)-dimensional vector-function \(u(x,y)\) of class \(C^2(\overline D)\) satisfying the boundary condition (52). From Green’s formula we obtain that equality (55) is fulfilled if and only if the equality holds

1. Differential Equations

\[ \int_{\Gamma}\left[ v\left( A_{20}\frac{\partial u}{\partial x}\cos(N,x) + A_{11}\frac{\partial u}{\partial y}\cos(N,x) + A_{22}\frac{\partial u}{\partial y}\cos(N,y) +\right.\right. \]
\[ \left. \left. {}+A_{10}u\cos(N,x)+A_{01}u\cos(N,y) \right) - \left( \frac{\partial}{\partial x}(vA_{20})\cos(N,x) +\right.\right. \]
\[ \left.\left. {}+\frac{\partial}{\partial x}(vA_{11})\cos(N,y) + \frac{\partial}{\partial y}(vA_{02})\cos(N,y) \right)u \right]\,ds=0. \tag{68} \]

Let the function \(u\in C^2(\overline D)\) and satisfy condition (52). Then, substituting the values of \(\dfrac{\partial u}{\partial x}\) and \(\dfrac{\partial u}{\partial y}\) from (60) and (61) into (68), we obtain

\[ \int_{\Gamma}\left(Z_1(v)g(s)+Z_2(v)u\right)\,ds=0, \tag{69} \]

where

\[ Z_1(v)=v\left(A_{20}\widetilde b_1\cos(N,x)+A_{11}\widetilde b_2\cos(N,x)+A_{02}\widetilde b_2\cos(N,y)\right), \]

\[ Z_2(v)=v\left(A_{20}\widetilde a_1\cos(N,x)+A_{11}\widetilde a_2\cos(N,x)+A_{02}\widetilde a_2\cos(N,y)+\right. \]
\[ \left. {}+A_{10}\cos(N,x)+A_{01}\cos(N,y) -\frac{\partial}{\partial x}(vA_{20})\cos(N,x) -\right. \]
\[ \left. {}-\frac{\partial}{\partial x}(vA_{11})\cos(N,y) -\frac{\partial}{\partial y}(vA_{02})\cos(N,y) \right). \]

It is easy to establish the following identity:

\[ \int_{\Gamma} Z_2(v(s))\left(\int_0^s K(s,\sigma)g(\sigma)\,d\sigma\right)\,ds = \int_{\Gamma}\int_{\Gamma} Z_2(v(\sigma))K(\sigma,s)g(s)\,ds\,d\sigma - \]
\[ {}-\int_{\Gamma}\left(\int_0^s Z_2(v(\sigma))K(\sigma,s)\,d\sigma\right)g(s)\,ds. \tag{70} \]

Here by \(Z_2(v(\sigma))\) and \(Z_2(v(s))\) we mean the value of \(Z_2(v)\) at the boundary points with arc abscissas \(\sigma\) and \(s\), respectively.

Substituting the value of \(u(x,y)\) from (67) into (69) and using identity (70), we obtain that equality (69) is satisfied for an arbitrary vector-function \(g(s)\) satisfying condition (66), and for arbitrary constants \(c_1,\ldots,c_{n-r}\), if and only if \(v(x,y)\) on \(\Gamma\) satisfies the conditions

\[ Z_1(v(s)) -\int_0^s Z_2(v(\sigma))K(\sigma,s)\,d\sigma + \int_{\Gamma} Z_2(v(\sigma))(K(\sigma,s)+K_1(\sigma,s))\,d\sigma = \]
\[ =\sum_{k=1}^{n-r} d_k\left(c_k^*K(l,s)\right), \tag{71} \]

\[ \int_{\Gamma} Z_2(v(s))\widetilde u_i(s)\,ds=0, \qquad i=1,2,\ldots,n-r, \tag{72} \]

where \(d_1,\ldots,d_{n-r}\) are constants.

We note that the \(n\)-dimensional vector-functions \(Z_1(v(s))\), \(Z_2(v(\sigma))\), and the \(n\)-dimensional constant vectors \(c_k^*\) in formula (71) are regarded as rows. Conditions (71), (72) are the adjoint boundary condition to boundary condition (52) with respect to system (51).

In the case of a multiply connected domain, the adjoint boundary condition is constructed analogously, with the only difference that equation (61) must in this case be solved separately for each contour \(\Gamma_k\) \((k=0,1,2,\ldots,m)\).

For example, the adjoint boundary condition to the boundary condition

\[ \left(a(s)\frac{\partial u}{\partial N}+b(s)\frac{\partial u}{\partial s}+c(s)u\right)\bigg|_{\Gamma}=0 \qquad \left(a^2(s)+b^2(s)=1\right) \]

with respect to the Poisson equation

\[ \frac{\partial^2 u}{\partial x^2}+\frac{\partial^2 u}{\partial y^2}=h \]

has the following form:

\[ b(s)v(s)-a(s)\psi(s)\int_0^s \left[ \frac{\partial v(\sigma)}{\partial n} +\bigl(c(\sigma)+k b(\sigma)\bigr)a(\sigma)v(\sigma) \right]\frac{d\sigma}{\psi(\sigma)} = \]

\[ =d_1 a(s)\psi(s), \]

if \(a(s)\neq 0\) and \(a(0)\neq 0\);

\[ v\big|_{\Gamma}=0,\quad \text{if } a\equiv 0,\quad \int_{\Gamma} c\,ds\neq 0; \]

\[ v\big|_{\Gamma}=0,\quad \int_{\Gamma}\psi_1(s)\frac{\partial v}{\partial N}\,ds=0,\quad \text{if } a\equiv 0,\quad \int_{\Gamma} c\,ds=0; \]

here \(d_1\) is a constant number; \(N\) is the inner normal to the contour \(\Gamma\),

\[ \psi(s)=e^{\int_0^s (b(\sigma)c(\sigma)-ka^2(\sigma))\,d\sigma}, \qquad k=\left(\int_{\Gamma}a^2\,ds\right)^{-1}\cdot \int_{\Gamma}bc\,ds, \]

\[ \psi_1(s)=e^{-\int_0^s c(\sigma)\,d\sigma}. \]

We note that if \(a\not\equiv 0\) on \(\Gamma\), then we can always measure the length of the arc \(s\) from a point \((x_0,y_0)\in\Gamma\) where \(a(x_0,y_0)\neq 0\), i.e. \(a(0)\neq 0\).

Remark 2. Equality (68) holds for any vector-functions \(u(x,y)\) and \(v(x,y)\) of the class \(C^1(\overline{D})\), satisfying respectively the boundary conditions (52) and (71), (72).

Let us return to the study of problem (1), (2).

In this paragraph we shall assume that the matrices \(a(z)\) and \(b(z)\) of boundary condition (2) satisfy condition (57).

We note that the solution of problem (1), (2) can always be reduced to the solution of an inhomogeneous equation (1) with homogeneous boundary conditions, i.e. to the solution of a system of the form

\[ L(u)\equiv A\frac{\partial^2 u}{\partial x^2}+2B\frac{\partial^2 u}{\partial x\partial y}+C\frac{\partial^2 u}{\partial y^2}=h(x,y) \tag{73} \]

with boundary conditions

\[ a(z)\frac{\partial u}{\partial x}+b(z)\frac{\partial u}{\partial y}+c(z)u=0 \quad \text{on } \Gamma, \tag{74} \]

where \(h\in C_\alpha^{\,n-1}(\overline D)\).

In problem (73), (74), the same smoothness is required of \(a(z)\), \(b(z)\), \(c(z)\) and of the solution \(u(z)\) as in problem (1), (2). Therefore, for convenience of notation, we shall formulate the theorem obtained in § 1 in terms of the adjoint problem for problem (73), (74).

Theorem 2. If for problem (73), (74) conditions 1 and 2 are satisfied (see § 1), then the number of linearly independent solutions of the homogeneous problem (73), (74) and the number of linearly independent solutions of the homogeneous adjoint problem are finite; in order that problem (73), (74) have a solution, it is necessary and sufficient that the vector-function \(h(x,y)\) satisfy the condition

\[ \iint_D (h(x,y),v_i(x,y))\,dxdy=0, \qquad i=1,2,\ldots,k', \tag{75} \]

where \(v_1(x,y),\ldots,v_{k'}(x,y)\) is a complete system of linearly independent solutions of the homogeneous adjoint problem.

Proof. The assertion of the finiteness of the number of linearly independent solutions of the homogeneous problem (73), (74) is contained in Theorem 1. The proof of the necessity of condition (75) for the solvability of problem (73), (74) is trivial. We shall prove the sufficiency of condition (75).

Introduce the notation

\[ W(x,y)=\iint_D v(x,y,\xi,\eta)h(\xi,\eta)\,d\xi d\eta, \]

where

\[ v(x,y,\xi,\eta)=\frac{1}{2\pi^2}\operatorname{Re}\int_\gamma \Delta(\lambda)\ln(\xi-x+\lambda(\eta-y))\,d\lambda; \]

\(\Delta(\lambda)\) is the matrix inverse to the matrix \(A+2B\lambda+C\lambda^2\), and \(\gamma\) is a contour in the half-plane \(\operatorname{Im}\lambda>0\), enclosing all roots of the polynomial \(\det(A+2B\lambda+C\lambda^2)\) lying in this half-plane. The function \(W(x,y)\) is a particular solution of system (73). By means of the substitution \(u=\omega(x,y)+W(x,y)\), problem (73), (74) is reduced to problem (1), (2). Hence, by virtue of the equivalent reduction of problem (1), (2) to a system of singular integral equations of normal type (see § 1), we obtain the following assertion: if the conditions of Theorem 2 are satisfied, then in order that problem (73), (74) have a solution, it is necessary and sufficient that the vector-function \(h(x,y)\) satisfy a finite number of conditions of the form

\[ \iint_D (\widetilde v_i(x,y),h(x,y))\,dxdy=0, \tag{76} \]

where \(\widetilde v_i(x,y)\in C_\alpha^n(\overline D)\).

We shall now prove that \(\widetilde v_i(x,y)\) is a solution of the adjoint problem. This will complete the proof of Theorem 2.

Let \(u(x,y)\) be any vector function of the class \(C_\alpha^n(\overline D)\) satisfying the boundary condition

\[ u\big|_\Gamma=0,\qquad \frac{\partial u}{\partial N}\bigg|_\Gamma=0. \tag{77} \]

Then the equality

\[ \iint_D (Z(u),\tilde v_i)\,dxdy = \iint_D (u,Z^*(\tilde v_i))\,dxdy \tag{78} \]

holds.

If we take \(h=Z(u)\), then, obviously, for \(h(x,y)\) problem (73), (74) has a solution. Consequently, according to (76), the left-hand side of equality (78) is equal to zero. Therefore the right-hand side of equality (78) is equal to zero for any vector function \(u\in C_\alpha^n(\overline D)\) satisfying condition (77). Hence,

\[ Z^*(\tilde v_i(x,y))=0. \tag{79} \]

Now let \(u(x,y)\) be any vector function of the class \(C_\alpha^n(\overline D)\) satisfying the boundary condition (74). It is shown analogously that the left-hand side of (78) is equal to zero. Consequently, in view of (79), equality (78) holds for any vector function \(u(x,y)\) of the class \(C_\alpha^n(\overline D)\) satisfying condition (74). It follows that (78) holds for any \(u(x,y)\in C^2(\overline D)\) satisfying condition (74). Consequently, by the definition of the adjoint boundary condition, we obtain that \(\tilde v_i\) satisfies the adjoint boundary condition.

§ 3. On the normal solvability of problem (51), (52)

For problem (51), (52) the following holds.

Theorem 3. For the solvability of problem (51), (52) it is necessary that the vector function \(h(x,y)\) satisfy the condition

\[ \iint_D (v,h)\,dxdy=0, \tag{80} \]

where \(v\) is any solution of the homogeneous adjoint problem.

This theorem is proved trivially (only in the case when \(k=1\), one must use Remark 2).

Definition 2. We shall call problem (51), (52) normally solvable if condition (80) is a necessary and sufficient condition for the solvability of this problem.

From this definition there follows immediately

Corollary 2. If problem (51), (52) is normally solvable and has solutions for the vector functions \(h_m(x,y)\) \((m=1,2,\ldots)\), then it has solutions \(u\) for the vector function \(h(x,y)\in C_\alpha^{k-1}(\overline D)\), which is the limit of the sequence of functions \(h_m(x,y)\) in the metric \(L_2\).

The following holds.

Theorem 4. If there exists a set of vector functions \(M\), which consists of a finite or infinite number of vector functions \(v_i\in C^2(D)\cap C_\alpha^k(\overline D)\), and such that, for the solvability of problem (51), (52), it is necessary and sufficient that the vector function \(h(x,y)\) satisfy the condition

\[ \iint_D (h(x,y), v_i(x,y))\,dxdy = 0 \]

for every \(v_i(x,y)\) from the set \(M\), then problem (51), (52) is normally solvable.

Theorem 4 is proved analogously to Theorem 2.

To formulate the next theorem we introduce some notation. If \(M\) is a certain class of vector-functions, then by \(\overline{M}\) we denote the closure of the set \(M\) in the metric \(L_2\). Denote by \(M(Z)\) the set of all possible values of the operator \(Z(u)\), when \(u(x,y)\) ranges over the class \(C^2(D)\cap C_\alpha^k(\overline D)\) and satisfies the boundary condition (52). By \(N\) we denote the orthogonal complement of the set
\[ M(Z)\cap C_\alpha^{k-1}(\overline D) \]
in \(L_2\).

Theorem 5. For normal solvability of problem (51), (52), it is sufficient that
\[ M(Z)\cap C_\alpha^{k-1}(\overline D) = \overline{M(Z)\cap C_\alpha^{k-1}(\overline D)} \cap C_\alpha^{k-1}(\overline D). \tag{81} \]

\[ N=\overline{N\cap C^2(D)\cap C_\alpha^k(\overline D)}. \tag{82} \]

If the set \(N\) is a finite-dimensional subspace, then conditions (81), (82) are necessary and sufficient for normal solvability of problem (51), (52).

Proof. Suppose that conditions (81), (82) are fulfilled. Then, in order that for a function \(h(x,y)\in C_\alpha^{k-1}(\overline D)\) problem (51), (52) have a solution, it is necessary and sufficient that the vector-function \(h(x,y)\) satisfy the condition
\[ \iint_D (v(x,y), h(x,y))\,dxdy = 0, \]
where \(v\) is any function from the set \(N\cap C^2(D)\cap C_\alpha^k(\overline D)\). Consequently, by Theorem 4, problem (51), (52) is normally solvable.

Let the set \(N\) be finite-dimensional, i.e., let there exist in \(N\) a finite number of linearly independent vector-functions. We shall prove the necessity of conditions (81), (82) for normal solvability of problem (51), (52). The necessity of condition (81) follows from Corollary 2. Denote by \(M'\) the set of solutions of the homogeneous adjoint problem. From (80) we obtain that \(M'\subset N\). Consequently, the set \(M'\) is finite-dimensional. Let problem (34), (35) be normally solvable and let \(v_1,\ldots,v_k\) be an orthonormal basis in \(M\). By the definition of the adjoint problem, \(v_i\in C^2(D)\cap C_\alpha^k(\overline D)\). We shall prove that \(N=M'\), whereby the necessity of condition (82) will also be proved. Suppose \(N\ne M'\). Then there exists a nonzero vector-function \(v_0\in N\) which is orthogonal to the set \(M'\). It is clear that there always exists a vector-function \(h_0(x,y)\in C_\alpha^{k-1}(\overline D)\) such that, for it,
\[ \iint_D (h_0(x,y), v_0(x,y))\,dxdy \ne 0. \tag{83} \]

Consider the vector-function

\[ h_1(x,y)=h_0(x,y)-\sum_{i=1}^{k}\left(\iint\limits_D \bigl(h_0(x,y),v_i(x,y)\bigr)\,dxdy\right)v_i(x,y). \]

It satisfies condition (80); therefore, for it the problem (51), (52) is solvable. On the other hand, \(h_1(x,y)\) satisfies inequality (83). Consequently, from the definition of the set \(N\) it follows that for \(h_1\) this problem has no solution. From the contradiction obtained it follows that \(N=M'\). The theorem is proved.

The problem adjoint to the Dirichlet problem for system (51) is the Dirichlet problem for the adjoint system. Theorems 3, 4, and 5 are also valid for the Dirichlet problem (i.e., for problem (51), (52), when (52) has the form \(\left.u\right|_\Gamma=0\)).

(Continued in the next issue)

Received by the editors
September 7, 1965

Institute of Mathematics
of the Siberian Branch of the Academy of Sciences of the USSR