N. E. TOVMASYAN

SOME BOUNDARY-VALUE PROBLEMS FOR SYSTEMS OF ELLIPTIC EQUATIONS OF SECOND ORDER THAT DO NOT SATISFY THE CONDITION OF YA. B. LOPATINSKII

(Presented by Academician M. A. Lavrent’ev on 10 VIII 1964)

1. In the present paper the following problem is considered:
In a bounded plane simply connected domain \(D\) with boundary \(\Gamma\), it is required to find a regular solution of the elliptic system

\[ A\frac{\partial^2 u}{\partial x^2}+2B\frac{\partial^2 u}{\partial x\partial y}+C\frac{\partial^2 u}{\partial y^2}=0, \tag{1} \]

belonging to the class \(C_\alpha^2(\overline D)\) and satisfying the boundary condition

\[ au_x+bu_y+cu=f \quad \text{on } \Gamma, \tag{2} \]

where \(u=(u_1,u_2)\) is the unknown vector, \(f=(f_1,f_2)\) is a given real vector on \(\Gamma\); \(A,B,C,a,b,c\) are real constant square matrices of second order.

Let \(z=t(s)\) \((z=x+iy)\) be a parametric equation of the boundary \(\Gamma\), where \(s\) is arc length. It is assumed that \(f\in C_\alpha^2(\Gamma)\), \(t(s)\in C_\alpha^3(\Gamma)\).

In paper \((^1)\) it was proved that the Ya. B. Lopatinskii condition ensures the Noether property of problem (1)—(2). In the present paper a more general condition is indicated under which problem (1)—(2) in any simply connected domain is Noetherian. If this condition is not fulfilled, then one can always indicate a domain with an arbitrarily smooth boundary where problem (1)—(2) is not Noetherian even in the case when \(f\) is an infinitely differentiable function.

We shall call problem (1)—(2) Noetherian if the homogeneous problem (1)—(2) has a finite number of linearly independent solutions, and the inhomogeneous problem is solvable when the function \(f\) is subject to a finite number of orthogonality conditions.

2. We first consider the case where the characteristic equation

\[ \left|A+2B\lambda+C\lambda^2\right|=0 \tag{3} \]

has multiple roots. Let \(\lambda_0\) be a multiple root of equation (3) with positive imaginary part. Without loss of generality we shall assume that \(\lambda_0=i\). This can always be achieved by means of a transformation of the independent variables of the form \(X=x+\operatorname{Re}\lambda_0 y,\ Y=\operatorname{Im}\lambda_0 y\).

In this case the general solution of system (1) in the domain \(D\) is given by the formula (see \((^2)\), p. 64)

\[ u=\operatorname{Re}\left[\alpha_1\varphi_1(z)+\alpha_2\left(\varphi_2(z)+kz\overline{\varphi_1'(z)}\right)\right], \tag{4} \]

where \(\varphi_1(z)\) and \(\varphi_2(z)\) are arbitrary analytic functions in the domain \(D\), with \(k=1\), while \(\alpha_1\) and \(\alpha_2\) \((\alpha_2\ne 0)\) are two-dimensional constant vectors that are particular solutions of the system of equations

\[ (A+2Bi-C)\alpha_2=0,\qquad (A+2Bi-C)\alpha_1+2(A+C)\alpha_2=0. \tag{5} \]

if the rank of the matrix \(A+2Bi-C\) is equal to 1, and \(k=0\), \(\alpha_1=(1,0)\), \(\alpha_2=(0,1)\), if all elements of the matrix \(A+2Bi-C\) are zero.

We note that the system of equations (5) always has solutions \(\alpha_1\) and \(\alpha_2\), where \(\alpha_2\ne 0\). Suppose that the origin of coordinates lies inside the domain \(D\). On the functions \(\varphi_1(z)\) and \(\varphi_2(z)\) one may impose the conditions
\[ \operatorname{Im}\bigl[(\alpha_1-2k\alpha_2)\varphi_1(0)+\alpha_2\varphi_2(0)\bigr]=0. \]
Under these conditions the analytic functions \(\varphi_1\) and \(\varphi_2\) are determined by \(u(x,y)\) uniquely.

Substituting the general solution (4) into the boundary condition (2), we obtain
\[ \operatorname{Re}\bigl[\beta_1\varphi_1(z)+\beta_2(\varphi_2(z)+k\bar z\varphi_1'(z))+\beta_3\varphi_1'(z)+ \]
\[ +\beta_4(\varphi_2'(z)+k\bar z\varphi_1''(z))\bigr]=f \quad\text{on }\Gamma, \tag{6} \]
where
\[ \beta_1=c\alpha_1,\quad \beta_2=c\alpha_2,\quad \beta_3=(a+bi)\alpha_1+k(a-bi)\alpha_2,\quad \beta_4=(a+bi)\alpha_2. \]

Let one of the following conditions be satisfied:

a) \(\beta_3=0,\ \beta_4=0\); the vectors \(\beta_1\) and \(\beta_2\) are linearly independent.

b) \(\beta_4=0,\ \beta_3\ne 0\); the vectors \(\beta_2\) and \(\beta_3\) are linearly independent.

c) \(\beta_4=0,\ \beta_3\ne 0\); the vectors \(\beta_2\) and \(\beta_3\) are linearly dependent, and the vectors \(\beta_1\) and \(\beta_2\) are linearly independent.

d) \(\beta_4\ne 0,\ k=0\); the vectors \(\beta_3\) and \(\beta_4\) are linearly dependent, \(\beta_2\) and \(\beta_4\) are linearly independent; \(|k_2-l_1|+|l_2|\ne 0\), where \(k_2,l_1\), and \(l_2\) are numbers determined from the equations
\[ \beta_3=k_2\beta_4,\qquad \beta_1=l_1\beta_2+l_2\beta_4. \tag{7} \]

e) \(\beta_4\ne 0,\ k=1\); the vectors \(\beta_3\) and \(\beta_4\) are linearly dependent, \(\beta_2\) and \(\beta_4\) are linearly independent; \(|k_2-l_1|\ne 0\), where \(k_2\) and \(l_1\) are determined from (7).

f) \(\beta_4\ne 0\), \(\beta_3\) and \(\beta_4\) are linearly dependent, \(\beta_1\) and \(\beta_4\) are linearly independent, \(\beta_2\) and \(\beta_4\) are linearly dependent.

g) \(\beta_3\) and \(\beta_4\) are linearly independent.

When any one of these conditions is fulfilled, problem (1)—(2) is reduced to an equivalent singular integral equation of normal type, if one uses, respectively for cases a)—g), the integral representations of the analytic functions \(\varphi_1(z)\) and \(\varphi_2(z)\) in the form
\[ \text{1)}\qquad \varphi_1(z)=\frac{1}{\pi i}\int_\Gamma \frac{\mu_1(t)\,dt}{t-z}+ic_1, \]
\[ \varphi_2(z)=\frac{1}{\pi i}\int_\Gamma \frac{\mu_2(t)-k\bar t\,\mu_1'(t)}{t-z}\,dt+ic_2. \]

Here and in what follows \(\mu_1(t)\) and \(\mu_2(t)\) are real functions; \(c_1\) and \(c_2\) are real constants, \(i\) is the imaginary unit;
\[ \mu'(t)=\frac{d\mu(t)}{dt},\qquad t=\xi+i\eta\in\Gamma,\qquad \bar t=\xi-i\eta,\qquad \bar t'=\frac{d\bar t}{dt},\qquad d_j=\int_\Gamma \mu_j(t)\,ds. \]

\[ \text{2)}\qquad \varphi_1(z)=\int_\Gamma \mu_1(t)\ln\left(1-\frac{z}{t}\right)\,ds+d_1+ic_1, \]
\[ \varphi_2(z)=\int_\Gamma \frac{\mu_2(t)+k\bar t\,\mu_1(t)\,\bar t_s'}{t-z}\,dt+c_2. \]

\[ \text{3)}\qquad \varphi_1(z)=\frac{1}{\pi i}\int_\Gamma \frac{\mu_1(t)\,dt}{t-z}+ic_1, \]
\[ \varphi_2(z)=\frac{1}{\pi i}\int_\Gamma \frac{\mu_2(t)-\mu_1'(t)(k_1+k\bar t)}{t-z}\,dt+ic_2, \]
where \(k_1\) is determined from the equality \(\alpha_3=k_1\alpha_2\), \(\bar t_s'=d\bar t/ds\).

4)

\[ l_1\varphi_1(z)+\varphi_2(z)=\frac{1}{\pi i}\int_\Gamma \frac{\mu_2(t)\,dt}{t-z}+ic_1, \]

\[ \varphi_1(z)=\int_\Gamma \left[\mu_1(t)+\frac{1}{\pi i\,(k_2-l_1)}\,\mu_2'(t)t_s'\right]\ln\left(1-\frac{z}{t}\right)ds+d_1+ic_2, \]

if \(k_2-l_1\ne 0\), and

\[ l_1\varphi_1(z)+\varphi_2(z)=\frac{1}{\pi i}\int_\Gamma \frac{\mu_2(t)\,dt}{t-z}+ic_1, \]

\[ l_2\varphi_1(z)=\frac{1}{\pi i}\int_\Gamma \frac{\bigl(\mu_1(t)-\mu_2'(t)\bigr)\,dt}{t-z}+ic_2, \]

if \(k_2-l_1=0\).

5)

\[ \varphi_1(z)=\int_\Gamma \frac{\mu_1(t)\,dt}{t-z}+c_1, \]

\[ \varphi_2(z)=\int_\Gamma \bigl(\mu_2(t)\bar t_s' + k\bar t\,\mu_1''(t)+k_2\mu_1'(t)+l_2\mu_1(t)\bigr)\times \]

\[ {}\times \ln\left(1-\frac{z}{t}\right)dt+d_2+ic_2, \]

where \(k_2\) and \(l_2\) are determined from (7).

6)

\[ \varphi_1(z)=\int_\Gamma \frac{\mu_1(t)\,dt}{t-z}+ic_1, \]

\[ \varphi_2(z)=\int_\Gamma \bigl(\mu_2(t)\bar t_s' + k\bar t\,\mu_1''(t)+k_2\mu_1'(t)\bigr)\ln\left(1-\frac{z}{t}\right)dt+d_2+ic_2. \]

7)

\[ \varphi_1(z)=\int_\Gamma \mu_1(t)\ln\left(1-\frac{z}{t}\right)ds+d_1+ic_1, \]

\[ \varphi_2(z)=\int_\Gamma \bigl(\mu_2(t)+k\bar t\,\mu_1'(t)\bigr)\ln\left(1-\frac{z}{t}\right)ds+d_2+ic_2. \]

In these integral representations the quantities \(\mu_1,\mu_2,c_1,c_2\) are determined uniquely by \(\varphi_1(z)\) and \(\varphi_2(z)\). The integral representations of analytic functions \(\varphi_1(z)\) and \(\varphi_2(z)\) used here are simple consequences of the integral representations given in (3). Analytic integral representations of analytic functions were used in the paper (4).

After such an equivalent reduction, by virtue of the known theorems of the theory of singular integral equations (see (3)), we obtain:

Theorem 1. If the characteristic equation (3) has a multiple root \(i\) and one of the conditions a)—ж) is satisfied, then problem (1)—(2) is Noetherian. The index of problem (1)—(2) is equal to zero if one of the conditions a), в) is satisfied; is equal to 2 if one of the conditions б), е) is satisfied; is equal to 4 if condition ж) is satisfied. If condition г) is satisfied, the index of problem (1)—(2) is equal to zero if \(k_2-l_1=0\), and is equal to 2 if \(k_2\ne l_1\). If condition д) is satisfied, the index of problem (1)—(2) is equal to 2 if \(|k_2-l_1|>1\), and is equal to 6 if \(|k_2-l_1|<1\).

Let us note that the Ya. B. Lopatinskii condition is equivalent to condition ж).

1. Let the characteristic equation (3) have simple roots. Denote by \(\lambda_1\) and \(\lambda_2\) the roots of this equation with positive imagi-

parts. The general solution of system (1) in this case can be written in the form (see (2))

\[ u=\operatorname{Re}\left(\gamma_1\varphi_1(z_1)+\gamma_2\varphi_2(z_2)\right), \tag{8} \]

where \(\gamma_1\) and \(\gamma_2\) are two-dimensional vectors which are nontrivial particular solutions of the equations

\[ A+2B\lambda_k+C\lambda_k^2\quad (k=1,2); \]

\(z_1=x+\lambda_1y,\ z_2=x+\lambda_2y\); \(\varphi_1(z)\) and \(\varphi_2(z)\) are arbitrary analytic functions, respectively, in the domains \(D_1\) and \(D_2\). The domains \(D_1\) and \(D_2\) are obtained from the domain \(D\) by means of the mappings \(\zeta=x+\lambda_1y\) and \(\zeta=x+\lambda_2y\).

The functions \(\varphi_1(z)\) and \(\varphi_2(z)\) at the zero point can be made subject to the conditions \(\operatorname{Re}(\lambda_1\varphi_1(0)+\lambda_2\varphi_2(0))\). Under these conditions \(\varphi_1(z)\) and \(\varphi_2(z)\) will be determined uniquely with the aid of \(u(x,y)\).

Substituting the general solution (8) into the boundary condition (2), we obtain

\[ \operatorname{Re}\left(\delta_1\varphi_1(z_1)+\delta_2\varphi_2(z_2)+\delta_3\varphi_1'(z)+\delta_4\varphi_2'(z)\right)=f \text{ on } \Gamma, \tag{9} \]

where \(\varphi_j'(z)=d\varphi_j(z)/dz,\ \delta_1=c\gamma_1,\ \delta_2=c\gamma_2,\ \delta_3=(a+\lambda_1b)\gamma_1,\ \delta_4=(a+\lambda_2b)\gamma_2\). If one of the vectors \(\delta_3\) and \(\delta_4\) is different from zero, we shall assume that \(\delta_3\ne0\) (this does not restrict generality).

Let one of the following conditions be fulfilled:

a\(^*\)) \(\delta_3=0,\ \delta_4=0\); the vectors \(\delta_1\) and \(\delta_2\) are linearly independent.

b\(^*\)) \(\delta_3\ne0,\ \delta_3\) and \(\delta_4\) are linearly dependent, while the vectors \(\delta_2\) and \(\delta_3\) are linearly independent,

\[ \left|(1-\lambda_1 i)l_3k_3-(1-\lambda_2 i)\right| \ne \left|(1+\lambda_1 i)l_3k_3-(1+\lambda_2 i)\right|, \]

where \(k_3\) and \(l_3\) are numbers determined from the equalities \(\delta_4=k_3\delta_3,\ \delta_1=l_3\delta_2+l_4\delta_3\).

c\(^*\)) \(\delta_3\ne0,\ \delta_3\) and \(\delta_4\) are linearly dependent, \(\delta_2\) and \(\delta_3\) are linearly dependent, \(\delta_1\) and \(\delta_3\) are linearly independent, \(|k_3|+|k_4|\ne0\), where \(k_4\) and \(k_3\) are determined from the equalities \(\delta_4=k_3\delta_3,\ \delta_2=k_4\delta_3\).

d\(^*\)) \(\delta_3\) and \(\delta_4\) are linearly independent.

Analogously to the case of multiple roots, in this case problem (1)—(2) also reduces to an equivalent one-dimensional singular integral equation of normal type, and the following theorem is proved.

Theorem 2. If the characteristic equation (3) has simple roots and one of the conditions a\(^*\))—d\(^*\)) is satisfied, then the homogeneous problem (1)—(2) in the class \(C_\alpha'(\overline D)\) has a finite number of linearly independent solutions, while the nonhomogeneous problem in the same class has a solution for \(f\in C_\alpha'(\Gamma)\), provided the function \(f\) is subject to a finite number of orthogonality conditions. The index of this problem is equal to zero if condition a\(^*\)) is satisfied and is equal to 4 if condition d\(^*\)) is satisfied. When condition b\(^*\)) is satisfied, the index of problem (1)—(2) is equal to 2 if the inequality

\[ \left|(1-\lambda_1 i)l_3k_3-(1-\lambda_2 i)\right| > \left|(1+\lambda_2 i)l_3k_3-(1+\lambda_2 i)\right| \]

holds, and is equal to 6 if the reverse inequality holds. When condition c\(^*\)) is satisfied, the index of problem (1)—(2) is equal to zero if \(k_3=0\), and is equal to 2 if \(k_3\ne0\). In this case the Ya. B. Lopatinskii condition is equivalent to condition d\(^*\)).

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

Received
4 VIII 1964

CITED LITERATURE

1. A. I. Volpert, Trudy Moskovsk. Matemat. Obshch., 10, 41 (1961).
2. A. V. Bitsadze, Equations of Mixed Type. Moscow, 1959.
3. N. I. Muskhelishvili, Singular Integral Equations. Moscow, 1946.
4. E. V. Zolotareva, DAN, 132, No. 4 (1960).