Preamble

DIFFERENTIAL EQUATIONS
FEBRUARY 1966, VOLUME II, NO. 2
UDC 517.946.9

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

In this paper, we consider a general boundary value problem for a system of $n$ second-order equations with constant coefficients of the form:
$$\begin{aligned} A \frac{\partial^2 u}{\partial x^2} + B \frac{\partial^2 u}{\partial x \partial y} + C \frac{\partial^2 u}{\partial y^2} = 0, \end{aligned}$$
where $A, B, C$ are constant square matrices of order $n$, and $u = (u_1, \dots, u_n)$ is the unknown vector.

The system is assumed to be elliptic in the sense of Petrovsky, meaning that the determinant of the characteristic matrix satisfies:
$$\det(A + B\lambda + C\lambda^2) \neq 0$$
for all real $\lambda$.

We investigate the problem in a simply connected domain $D$ bounded by a smooth closed curve $\Gamma$. The boundary conditions are specified as:
$$\left. \left( G \frac{\partial u}{\partial x} + H \frac{\partial u}{\partial y} + Q u \right) \right|_\Gamma = f(s),$$
where $G, H, Q$ are given constant matrices of size $n \times n$, and $f(s)$ is a given vector function of the arc length $s$ on $\Gamma$.

The objective of this study is to establish the conditions for the Noethericity (Fredholm property) of this boundary value problem and to calculate its index. By employing the method of singular integral equations, we reduce the problem to a system of functional equations on the boundary.

1. Reduction to a Canonical Form

To analyze the system, we first utilize the algebraic properties of the characteristic polynomial. Since the system is elliptic, the roots of the equation $\det(A + B\lambda + C\lambda^2) = 0$ are all non-real. Let these roots be denoted by $\lambda_k$ ($k = 1, \dots, 2n$). Without loss of generality, we assume that $n$ roots

§ 4. ON THE NORMAL SOLVABILITY OF THE DIRICHLET PROBLEM

For system (1), reference \cite{8} provides examples of elliptic systems for which the homogeneous Dirichlet problem in a disk possesses an infinite number of linearly independent solutions. This indicates that the Dirichlet problem for elliptic systems is, generally speaking, not Noetherian. However, the Dirichlet problem for system (1) may still be normally solvable, despite the fact that the homogeneous problem has infinitely many linearly independent solutions, as the definition of normal solvability does not require the number of linearly independent solutions to be finite. Consequently, the following questions naturally arise: 1. Is the Dirichlet problem for the elliptic system (1) always normally solvable? 2. Do there exist Dirichlet problems for elliptic systems that are normally solvable but not Noetherian? An answer to the second question is provided by the following example of a Dirichlet problem for an elliptic system (see \cite{8}). Find a solution to the elliptic equation, regular in the unit disk $|z| < 1$:

$$u = u_1 + i u_2$$

satisfying the boundary condition

$$u|_{\Gamma} = h(x, y),$$

where $h(x, y)$ is a given complex-valued function continuous in the closed domain. It was proven in \cite{8} that for this problem to be solvable, it is necessary and sufficient for the function $h(x, y)$ to satisfy a countable number of conditions of the form (81), where $(D)$. It follows from Theorem 4 that this problem is normally solvable, even though the homogeneous problem has an infinite number of linearly independent solutions. Therefore, for the elliptic system (84), the Dirichlet problem is normally solvable but not Noetherian. The answer to the first question is negative. To demonstrate this, consider the following example: find a solution to the elliptic system, regular in the unit disk $|z| < 1$ (the first part of this work can be found in Differential Equations, No. 1, 1966):

belonging to the class and satisfying the boundary condition

and $M = 0$, (86)

where $f$ is a given continuous function in $|z| < 1$, and $a$ is a constant. Equation (85) represents an elliptic system of two differential equations with real coefficients. We shall prove that the homogeneous problem (85), (86) possesses no non-trivial solutions, and that the problem itself is not normally solvable. The general solution of the homogeneous equation (85) takes the form:

$$u = \phi(x + iy) + \psi(x + 2iy)$$ (87)

where $\phi(z)$ and $\psi(z)$ are analytic functions in the disk $|z| < 1$ and in the ellipse $x^2 + \frac{y^2}{4} < 1$, respectively. Let $u(x, y)$ be a solution to the homogeneous problem (85), (86). Then, according to (86) and (87), $u(x, y)$ can be represented in the form (87), where the functions $\phi$ and $\psi$ satisfy the condition $\phi(x + iy) + \psi(x + 2iy) = 0$. Since $u \in C(D)$, it follows from (87) that $\phi(x + iy)$ and $\psi(x + 2iy)$ also belong to the class $C(D)$. Let us denote $\psi_1(z) = \psi(z)$. Since $\psi(z)$ is analytic in the ellipse (88), the function $\psi_1(z)$ is analytic with respect to $z$ in the annulus $\rho < |z| < 1$ and is continuous in the closed annulus $\rho \le |z| \le 1$. It is easily verified that

4 W H ' ( - 7 T ) '

According to $(89)$, we obtain $(2) = 0$ when $|z| = 1$. It follows that $\phi(z) + \psi(z) = 0$ when $\frac{1}{L} < |z| < L$. Consequently, the equality $(90)$ also holds for $\phi(z)$. By expanding the function $\phi(z)$ in a power series within the disk $|z| < 1$ and applying equality $(90)$, we find that $\phi(z) = c$ (where $c$ is a constant). From this and $(91)$, we obtain $\psi_1(z) = -c$. Since $\psi_1(z) = \psi(\frac{1}{z})$, it follows that $\psi(x + iy) = -c$. Substituting the determined values of the functions $\phi$ and $\psi$ into formula $(87)$, we conclude that $u(x, y) = 0$ in the domain.

Remark 3. Similarly, it can be proven that the homogeneous problem $(85)$, $(86)$ possesses only the trivial solution even when this solution

satisfies the boundary condition $(86)$ on an arbitrarily small part of the boundary, while on the remainder of the boundary, continuity of the desired solution is not even required. We shall now prove that problem $(85)$, $(86)$ is not normally solvable. According to Corollary 1, to prove this assertion, it is sufficient to show that for this problem, the function

3 ' - (z=x~iy) (92) 2(1 — ezf

has a solution if $0 < c < \frac{1}{3}$ and has no solution if $c = \frac{1}{3}$.

then it is easy to verify that the function $f(x) = \sqrt{x^2 + 3}$

$z = 2, \quad P = \sqrt{x^2 + 3}$,

$= P$

a r g ( : - M l 3i, p = a r g ( ^ - / - K ' 3 ) ) — < a , p < 2 * + — ) 2 1 2 j

is a solution to the problem (85), (86) for any $h(x, y)$ of the form (92), where $z = x + iy$ and $\bar{z} = x - iy$. Let $c = \dots$. We shall show that for $h(x, y)$ of the form (92), problem (85), (86) has no solution. Suppose, to the contrary, that problem (85), (86) for such an $h(x, y)$ has a solution $u_0(x, y)$. Then the function $v(x, y) = u(x, y) - u_0(x, y)$, given by formula (93), satisfies the homogeneous equation (85) in the domain $|z| < 1$ and the boundary condition (86) at all points of the boundary $|z| = 1$, except perhaps at the point $z = -1$. According to Remark 3, it follows that $u(x, y) = v(x, y) + u_0(x, y)$. However, since the function $u_0(x, y)$ is continuous at the point $z = -1$, while the function $u(x, y)$ has a singularity at this same point, equality (94) cannot hold. The resulting contradiction proves our assertion. From the examples provided, it is evident that the class of elliptic systems for which the Dirichlet problem is normally solvable is broader than the class of elliptic systems for which the problem is Noetherian, yet narrower than the entire class of elliptic systems. In [12], the following assertion was proved: if the characteristic equation (3) has an $n$-fold root $\lambda$, then for the Dirichlet problem for system (1) in a disk to be Noetherian, it is necessary and sufficient that system (1) be weakly coupled. This assertion remains valid regardless of the nature of the multiplicity of the roots in any finite domain with an analytical boundary and in the half-plane.

TOVMASYAN It should be noted that in a domain with an analytical boundary, the condition that the system be weakly coupled remains necessary for the Noetherian property of the Dirichlet problem even when the boundary data belong to the class of infinitely differentiable vector functions and the solution is sought in the class $C(\bar{D}) \cap C^2(D)$. A comparison of the results in the following section with the aforementioned assertion indicates that the class of elliptic systems for which the Dirichlet problem is normally solvable is much broader than the class of elliptic systems for which the Dirichlet problem is Noetherian. For the equation, the following theorem holds:

Theorem. For the Dirichlet problem of the system in a domain with an analytical boundary to be normally solvable, it is necessary and sufficient that the function $z(\zeta)$, which conformally maps the unit disk onto the domain, be rational.

This theorem implies that the normal solvability of the Dirichlet problem for strongly coupled systems depends very heavily on the configuration of the domain. Using the general solution of equation (84) (see [8]).

U = q(z )+~Z^(Z) + n ) d g d 4 / :

If $\phi(z)$ and $\psi(z)$ are arbitrary analytic functions in the domain, the theorem can be proven quite simply. We omit the proof here.

The Dirichlet Problem for the System in the Case of a Half-Plane and a Disk

In this section, we present a simple method for solving the Dirichlet problem for the system in the case of a half-plane and a disk. For the disk, we consider the Dirichlet problem only for systems where the characteristic equation has an $n$-fold root.

The Dirichlet problem for the half-plane is considered in the following formulation:

Problem. Find a regular solution to the system in the half-plane $y > 0$ that belongs to the class $C^0$ in any closed semi-disk $|z| \le R, y \ge 0$, and satisfies the following conditions:

u(x,0) = f(x), — oo < # < o o , • (96)

— represents the maximum order of the highest derivatives of the functions $u_j$ appearing in the general solution (10). Regarding the vector-function $f(x)$ defined on the axis, it is assumed to belong to the class $C^{(l)}$ on any finite interval, and $g(x) = f(1/x)$ belongs to the class $C^{(l)}$ in the neighborhood of the point $x=0$. In particular, finite functions satisfy these requirements. The following theorem holds:

Theorem. If the vectors $a_r$ ($r = 1, \dots, \nu$) appearing in the general solution of the system are linearly independent, then the homogeneous problem $A(f = 0)$ has only the trivial solution, and the non-homogeneous problem is always solvable. If, however, these vectors are linearly dependent, then the homogeneous problem

Problem A has an infinite number of linearly independent solutions; for the solvability of the non-homogeneous problem A, it is necessary and sufficient that the vector-function $f(x)$ satisfies an infinite number of conditions $(f, \psi_k) = 0$ ($k = 1, 2, \dots$), where $\psi_k$ is a set of linearly independent vectors. Here, $c_{rj}$ ($r = 1, \dots, \nu; j = 1, \dots, n$) are constant $n$-dimensional vectors that serve as linearly independent solutions to the algebraic system adjoint to the system $L u = 0$.

Proof. By substituting the general solution (10) into the boundary condition (9b), we obtain

k i

( S S V

w / r ( ^ ) ) , = o - f W - c , (99)

$\Phi(z)$ is an analytical function in the half-plane $\text{Im } z > 0$. From (10), it follows directly that the expression

$\Phi_j^{(r-1)}(x + \lambda_j y) + \gamma_j^{(1)} \Phi_j^{(r-2)}(x + \lambda_j y) + \dots + \gamma_j^{(r-1)} \Phi_j(x + \lambda_j y) y^{r-1}$

($r = 1, \dots, k_j$; $j = 1, \dots, N_0$)

can be expressed linearly through the first derivatives of the desired solution. Consequently, $\Phi(z)$ is analytic in any semi-disk $\text{Im } z > 0$, and the following estimates hold:
$|\Phi^{(r-1)}(z)| \le \text{const} \cdot \ln(2 + |z|), \quad \text{Im } z > 0$ (100)

( r = 1, kj, / = 1, v 0 ).

From equations (99) and (100), we obtain:
$$\int_{0}^{\infty} \Phi(t, z) \phi(t) \, dt = f(z) - c \tag{101}$$
where $c$ is a constant real vector. Let the vectors $\xi_i$ ($i = 1, \dots, n$; $j = 1, \dots, \nu$) be linearly independent. Then, by solving equation (101) with respect to $\phi_j(t)$ and substituting these solutions into (10), we obtain the desired solution to Problem L. To verify that this solution exhibits the required behavior at infinity, one must utilize the asymptotic estimate for large $|z|$:

$$|u(x, y)| \le \frac{C}{|z|^k} \quad (k = 0, 1, 2, \dots, k_0). \tag{102}$$

The estimate (102) follows from the behavior of the given vector function $f(z)$ at infinity. From (10) and (101), the uniqueness of the solution to Problem L follows. Now, let us assume the vectors $\xi_i$ ($i = 1, \dots, n$) are linearly independent. Let $r$ denote the number of linearly independent vectors $\xi_{ij}$ ($i = 1, \dots, n; j = 1, \dots, \nu$). In this case, equation (101) possesses a solution if and only if $f(z)$ satisfies the conditions:
$$f(t) - f(\infty) = 0 \quad (k = 1, \dots, n - r) \tag{103}$$
Condition (103) is equivalent to condition (97). When condition (97) is satisfied, the particular solution of system (101) takes the form:
$$\phi(t) = M^{-1} f(t) \tag{104}$$
where $\phi(t) = (\phi_1, \phi_2, \dots, \phi_\nu)^T$ and $M$ is a well-defined $n \times \nu$ constant matrix. Substituting (104) into (10), we obtain a particular solution to Problem L. Consequently, condition (97) is both necessary and sufficient for the solvability of Problem L.

The solution to the homogeneous Problem L is given by formula (10), where $\Phi(z)$ is replaced by $Q_i(z) + \Phi(z)$; here, $a_1, \dots, a_{n-r}$ are $n$-dimensional constant vectors that serve as linearly independent solutions to the system (98). The functions $Q_1(z), \dots, Q_{n-r}(z)$ are arbitrary analytical functions in the upper half-plane $y > 0$, chosen such that the solution $u(x, y)$ obtained via formula (10) maintains the required smoothness and behavior at infinity. In particular, $Q_i(z)$ can be taken in the form $Q_i(z) = (z - z_0)^{-k}$, where $z_0$ is an arbitrary fixed point in the lower half-plane and $k = 1, 2, \dots$. Consequently, in this case, the homogeneous Problem L possesses an infinite number of linearly independent solutions.

Remark:

3. Theorem 7 remains valid (only slightly changing)

The condition of the form (92) remains valid even if the half-plane $y > 0$ is replaced by any other half-plane in the $(x, y)$ plane. Let us now consider the Dirichlet problem for system (1) in the disk $|z| < 1$ under the following formulation: find a regular solution to the equation in the disk $D$ belonging to the class $C^1(\bar{D})$ and satisfying the boundary condition:
$$u|_{|z|=1} = f(z) \quad (105)$$
where $f(z)$ is a given vector function from the class $C^\alpha$. We shall assume that $\lambda$ is an $n$-fold root of the characteristic equation (3). Then the general solution (10) in the disk can be rewritten in the form:
$$u(x, y) = \text{Re} \left( \sum_{k=1}^n \delta_k \phi_k(z) + \sum_{k=2}^n \delta_k \sum_{j=1}^{k-1} \frac{(z-\bar{z})^{j}}{j!} \phi_k^{(j)}(z) \right) \quad (106)$$
where $\phi_k(z)$ are arbitrary analytic functions in the disk $|z| < 1$.

Substituting the general solution (106) into the boundary condition (105), we obtain:
$$\text{Re} \left( \sum_{j=1}^n \delta_j \psi_j(z) \right)_{|z|=1} = f(z) \quad (107)$$
where
$$\psi_j(z) = \phi_j(z) + \sum_{k=j+1}^n \frac{(z-\bar{z})^{k-j}}{(k-j)!} \phi_k^{(k-j)}(z) \quad (108)$$
($j = 1, 2, \dots, n$).

It is clear that $\psi_j(z)$ are analytic functions that may have poles at the point $z = 0$ of order no higher than $n-1$. First, we find the solutions $\psi_1, \dots, \psi_n$ of the boundary value problem (107) that are analytic in the disk $|z| < 1$ and have poles at $z = 0$ of order at most $n-1$. From (107), we have:
$$\sum_{j=1}^n \delta_j \psi_j(z) = \Phi(z) + \sum_{j=1}^{n-1} \frac{c_j}{z^j} \quad (109)$$
where $c_j$ ($j = 1, \dots, n-1$) are constant vectors, and the purely imaginary vector $c_n$ is defined accordingly. Let the number of linearly independent vectors $\delta_j$ be $r$. Then (109) has a solution in the considered class if and only if the following conditions are satisfied:
$$\text{Re} \int_{|z|=1} (f, \chi_k) ds = 0 \quad (110)$$
where $k = 1, \dots, n-r$, and $\chi_k$ are linearly independent solutions of the linear homogeneous algebraic system adjoint to the system:
$$\sum_{j=1}^n \delta_j x_j = 0 \quad (111)$$
When these conditions are met, the solution to system (109) is given by the formula:
$$\psi(z) = \mathbf{A} \Phi(z) + \sum a_j \omega_j + \sum_{k=1}^{n-1} \frac{g_k}{z^k} \quad (112)$$
where $\mathbf{A}$ is a well-defined constant $n$-dimensional matrix constructed using the vectors $\delta_1, \dots, \delta_n$; $\omega_j$ are linearly independent solutions of the algebraic system (111); and $g_k$ are arbitrary vectors. The condition can be written as $j = 1, \dots, p$ (113).

TOVMASYAN $(j=1, \dots, p)$ (114).

$(f_k = 1, n-r_0, j = 1, n-1)$.

The analytic functions $Q_i(z)$ appearing in (112) can be represented as $(j=1, \dots, p)$ (116), where $h = -k+1$ are arbitrary analytic functions in the domain $|z| < R$. Let the vector-function $\psi(z)$ and the vectors $g_i$ satisfy conditions (113)–(115). Then, by substituting the value of $\psi$ from (112) into (108) and solving (108) with respect to $(\phi_1(z), \dots, \phi_n(z))$, we find that this solution is regular at zero if and only if a finite number of conditions of the following form are satisfied:
$$\sum a_j M_j + \int (g_i, \Theta_i) dz = 0, \quad (117)$$
where $a_j$ are constant numbers, $g_i$ are constant vectors, and $\Theta_i$ are vector-functions. These quantities are fully determined. It is clear that part of the conditions (114), (115), and (117) can be satisfied using the vectors $g_i$ and the numbers $a_j$; the remaining part of the conditions can be replaced by conditions of the form:
$$(f, \chi_k) = d_k \quad (k = 1, 2, \dots, N). \quad (118)$$

Condition (110) together with (118) is necessary and sufficient for the solvability of the Dirichlet problem considered here in the disk. If conditions (110) and (118) are fulfilled, the solution to the problem is given by formula (106), where $\Phi(z) = (\phi_1(z), \dots, \phi_n(z))$ is an arbitrary solution of system (107); $\psi(z)$ is determined by (112); $Q$ is given by (116); $g_i$ are arbitrary vectors (where $g$ is a real vector); and $a_j$ are arbitrary numbers satisfying conditions (114), (115), and (117).

We note that the Dirichlet problem for weakly coupled systems in a disk was solved by a slightly different, yet similarly simple method in work \cite{12}. Using the results obtained, it is easy to prove that the Dirichlet problem for the inhomogeneous system (1) with homogeneous boundary data in the case of a half-plane or a disk is always normally solvable. The problems considered in this section are related to the works of A. Dzuraev \cite{A} and Ya. B. Lopatinskii \cite{L}.

\cite{A} A. Dzuraev, Doklady AN Tadzhikskoi SSR, vol. VII, no. 4, 1964.
\cite{L} Ya. B. Lopatinskii, Ukr. matem. zhurnal, no. 2, 1959.## § 5 Conclusion

The case where $n=2$ and the value represents a double root of the characteristic equation finds significant application in the plane theory of elasticity.

References

Bitsadze, A. V. Boundary value problems for systems of linear equations of elliptic type. Reports of the Academy of Sciences of the Georgian SSR, 5, No. 3, 1944.

Vekua, I. N. New Methods for Solving Elliptic Equations. Moscow, 1948.

Vishik, M. I. On strongly elliptic systems of differential equations. Matematicheskii Sbornik, 29(71), 3, 1951. Boyarskii, B. V. Doklady Akademii Nauk SSSR, No. 1, 1959. Volpert, A. I. On the index and normal solvability of boundary value problems for elliptic systems of differential equations in the plane. Transactions of the Moscow Mathematical Society.

Dzhuraev

A. Reports of the Academy of Sciences of the Tajik SSR, Vol. VII, No. 4, 1964. Lopatinskiy, Ya. B. Ukrainian Mathematical Journal, No. 2, 1959.

8. Bitsadze

References

Bitsadze, A. V. Equations of Mixed Type. Publishing House of the Academy of Sciences of the USSR, Moscow, 1959.

9. M u s k h e l i s h v i l i

Muskhelishvili, N. I. Singular Integral Equations. Moscow, 1946.

Faddeev, D. K., and Faddeeva, V. N. Computational Methods of Linear Algebra. Fizmatgiz, 1960.

11. S c h e c h t e r

Scheffer, M. General boundary value problems for elliptic partial differential equations. Communications on Pure and Applied Mathematics, vol. XII, no. 3, 1959.

Zolotareva, E. V. Doklady Akademii Nauk SSSR, no. 5, 1962.

Muskhelishvili, N. I. Some Basic Problems of the Mathematical Theory of Elasticity. Moscow, 1949.

Received by the editorial office on May 23, 1965.
Institute of Mathematics, Siberian Branch of the Academy of Sciences of the USSR.