Reports of the Academy of Sciences of the USSR
1960. Volume 132, No. 4

MATHEMATICS

E. V. ZOLOTAREVA

THE DIRICHLET PROBLEM FOR A CLASS OF ELLIPTIC SYSTEMS

(Presented by Academician I. N. Vekua on February 3, 1960)

It is known that the Dirichlet problem (the first boundary value problem) for linear systems of elliptic type in the sense of I. G. Petrovskii is, generally speaking, not a problem of Fredholm type \((^1)\). The Fredholm alternatives for the indicated problem always hold if the usual ellipticity requirement is strengthened by the condition of strong ellipticity \((^2)\). It is of interest to study the Dirichlet problem for elliptic systems without imposing the condition of strong ellipticity.

Let there be given a system of linear partial differential equations of the second order of the form

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

where \(A=\|A_{ik}\|\), \(B=\|B_{ik}\|\), \(C=\|C_{ik}\|\) are given real constant square matrices of order \(n\); \(u=(u_1,u_2,\ldots,u_n)\) is the vector sought.

The system (1) is called elliptic in the sense of I. G. Petrovskii if all \(2n\) roots of the characteristic equation

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

are complex (non-real).

Denote by \(\lambda_1,\ldots,\lambda_\mu,\ \bar{\lambda}_1,\ldots,\bar{\lambda}_\mu\) the roots of equation (2), and by \(k_1,\ldots,k_\mu\) the multiplicities of these roots.

The general representation of regular solutions of system (1) in a finite simply connected domain \(D\) has the form \((^3)\)

\[ u=\operatorname{Re}\sum_{j=1}^{\mu}\sum_{l=1}^{k_j}\sum_{m=0}^{l-1} C_{lm}^{(j)} \bar{z}_j^{\,m}\varphi_{jl}^{(m)}(z_j), \tag{3} \]

where the \(\varphi_{jl}\) are arbitrary holomorphic functions in the domain \(D\) of the variables \(z_j=x+\lambda_j y\), respectively; the upper index \(m\) of the function \(\varphi_{jl}\) indicates the order of derivative with respect to \(z_j\), and \(C_{lm}^{(j)}\) are completely determined constant vectors, expressible solely in terms of the coefficients of system (1), with the determination of \(C_{lm}^{(j)}\) requiring the solution of linear algebraic systems whose coefficients are expressed through the coefficients of system (1).

The Dirichlet problem for system (1) consists in determining a solution of this system, regular in the domain \(D\), continuous up to the contour \(\Gamma\) of the domain \(D\), and satisfying the condition

\[ u=f \quad \text{on } \Gamma, \tag{4} \]

where \(f=(f_1,\ldots,f_n)\) is a given vector.

In the present note we confine ourselves to the study of the Dirichlet problem for system (1) in the case \(n=2\) in a circular domain, when equation (2) has a multiple root.

Let \(\lambda\) be a double root of equation (2). Without loss of generality one may assume that \(\lambda=i\). This can always be achieved by a nonsingular affine transformation of the independent variables; however, if before the change of variables the domain \(D\) was the disk \(|z|<1\), then after the transformation it will, generally speaking, become an ellipse.

Thus, we shall assume that \(\lambda=i\) is a double root of equation (2), and that the domain \(D\) coincides with the disk \(|z|<1\). Suppose also that the vector \(f=(f_1,f_2)\) prescribed on the circumference \(\Gamma\) of the disk \(D\) satisfies the Hölder condition.

In the case under consideration, the general representation of the solutions of system (1) regular in the domain \(D\), in view of (3), has the form

\[ u=\operatorname{Re}\,[a\Phi(z)+a_1\Psi(z)+a_2\bar z\Psi'(z)], \tag{5} \]

where \(\Phi(z)\) and \(\Psi(z)\) are functions of the variable \(z=x+iy\), holomorphic in \(D\), and the vectors \(a,a_1,a_2\) are solutions of the system

\[ (A+2Bi-C)a=0,\qquad (A+2Bi-C)a_2=0, \]
\[ (A+2Bi-C)a_1+2(A+C)a_2=0. \tag{6} \]

The rank \(r\) of the matrix \(A+2Bi-C\) is either 1 or 0. In the case \(r=1\), choosing a definite (nonzero) value for the vector \(a\), we shall have \(a_2=\mu a\), where \(\mu\) is a scalar. For the third group of equations (6), with respect to the vector \(a_1\), the solvability conditions will be satisfied, and we can write \(a_1=\nu a+\mu b\), where \(\mu b\) is a particular solution of the nonhomogeneous system of this same group of equations.

Consequently, the general representation (5) can be rewritten in the form

\[ u=\operatorname{Re}\,[a\varphi(z)+b\psi(z)+a\bar z\psi'(z)], \tag{7} \]

where \(\Phi(z)+\nu\Psi(z)\equiv\varphi(z)\), \(\mu\Psi(z)\equiv\psi(z)\).

If \(r=0\), then obviously \(B=0\), \(A=C\), and consequently system (1) has the form \(A(u_{xx}+u_{yy})=0\). But \(\det A\ne0\). Hence we conclude that system (1) is split: \(\Delta u_i=0\) \((i=1,2)\), where \(\Delta\) is the Laplace operator. Therefore we immediately conclude that for \(r=0\) the Dirichlet problem for system (1) is always solvable, and in a unique way.

It remains to consider the case \(r=1\). We shall start from the general representation (7). Equating the expression for \(u(x,y)\) from (7) on the circumference \(\Gamma\) to the function \(f\), we readily obtain the relation

\[ a\varphi(z)+b\psi(z)+a\frac{\psi'(z)-\psi'(0)}{z} +\overline{a\varphi(0)}+\overline{b\psi(0)} +\overline{a\psi''(0)}+\overline{a\psi'(0)}z = \]
\[ =\frac{1}{\pi i}\int \frac{f(t)\,dt}{t-z}. \tag{8} \]

Following (3), we shall call system (1) weakly coupled if the vectors \(a\) and \(b\) are linearly independent. Otherwise system (1) will be called strongly coupled. In view of the fact that

\[ a\varphi(0)+b\psi(0)+a\psi''(0) +\overline{a\varphi(0)}+\overline{b\psi(0)}+\overline{a\psi''(0)} =\frac{1}{\pi i}\int\frac{f\,dt}{t}, \]
\[ a\varphi'(0)+b\psi'(0)+\frac{a}{2}\psi'''(0) +\overline{a\psi'(0)} =\frac{1}{\pi i}\int\frac{f\,dt}{t^2}, \tag{9} \]

relation (8) can be rewritten in the form

\[ a[\varphi(z)-\varphi(0)-\varphi'(0)z] +b[\psi(z)-\psi(0)-\psi'(0)z]+ \]
\[ +a\frac{\psi'(z)-\psi'(0)-\psi''(0)z-\frac12\psi'''(0)z^2}{z} = \frac{1}{\pi i}\int_{\Gamma} \left(\frac{1}{t-z}-\frac{1}{t}-\frac{z}{t^2}\right)f(t)\,dt. \tag{10} \]

In the case of a weakly coupled system (1), for the homogeneous Dirichlet problem \((f=0)\), from (10) we obtain

\[ \psi(z)=\psi(0)+\psi'(0)z,\qquad \varphi(z)=\varphi(0)+\varphi'(0)z. \]

Hence it follows that the homogeneous Dirichlet problem under consideration has only the trivial solution

\[ u_0(x,y)=\operatorname{Re}\{a[\varphi(0)+\varphi'(0)z]+b[\psi(0)+\psi'(0)z]+\bar a z\psi'(0)\}=0. \]

Using formula (10) again, we conclude that the inhomogeneous Dirichlet problem always has a unique solution, and it is written explicitly in the form

\[ \begin{aligned} u(x,y)=(z\bar z-1)\operatorname{Re}\Biggl\{ &aM\frac{1}{\pi i}\int_{\Gamma}\frac{f_1(t)(2t-z)}{t^2(t-z)^2}\,dt +\frac{aN}{\pi i}\int_{\Gamma}\frac{f_2(t)(2t-z)}{t^2(t-z)^2}\,dt \Biggr\}\\ &+\operatorname{Re}\frac{1}{\pi i}\int_{\Gamma}\frac{f\,dt}{t-z} -\frac{1}{2\pi i}\int_{\Gamma}\frac{f(t)\,dt}{t}, \end{aligned} \tag{11} \]

where

\[ M=-\frac{a_2}{a_1b_2-a_2b_1},\qquad N=\frac{a_1}{a_1b_2-a_2b_1}. \]

If system (1) is strongly coupled, we have \(b=\mu a\), where \(\mu\) is a scalar parameter. In this case, by virtue of (10), the inhomogeneous Dirichlet problem will be solvable only under the identical fulfillment of the condition

\[ \int_{\Gamma}\left(\frac{1}{t-z}-\frac{1}{t}-\frac{z}{t^2}\right)(a_2f_1(t)-a_1f_2(t))\,dt=0. \tag{12} \]

Identity (12) is obviously equivalent to the conditions

\[ \int_{\Gamma}\bigl(\bar a_2 f_1(t)-\bar a_1 f_2(t)\bigr)t^k\,dt=0,\qquad k=1,2,\ldots, \tag{13} \]

and, if conditions (13) are satisfied, then the solution of the inhomogeneous Dirichlet problem exists and is written explicitly as

\[ u(x,y)=(z\bar z-1)\operatorname{Re}a\frac{\psi'(z)-\psi'(0)}{z} +\operatorname{Re}\frac{1}{\pi i}\int_{\Gamma}\frac{f(t)\,dt}{t-z} -\frac{1}{2\pi i}\int_{\Gamma}\frac{f(t)\,dt}{t}. \]

Here the expression

\[ u_0(x,y)=(z\bar z-1)\operatorname{Re}a\frac{\psi'(z)-\psi'(0)}{z} \]

for an arbitrary holomorphic function \(\psi(z)\) is a solution of the homogeneous Dirichlet problem.

It is not difficult to show that, for weak coupling of system (1), in the case when \(\lambda=i\) is a double root and \(n=2\), the necessary and sufficient condition is

\[ \det|A+C|\ne 0. \tag{14} \]

In the case when, instead of \(i\), the double root is \(\lambda\), condition (14) takes the form

\[ \det|A+B\lambda+\bar B\bar\lambda+C\lambda\bar\lambda|\ne 0. \]

We have shown that in any finite simply connected domain the weak coupling of system (1) is a condition necessary and sufficient for the Fredholm property of the Dirichlet problem.

V. A. Steklov Mathematical Institute
Academy of Sciences of the USSR

Received
30 I 1960

REFERENCES

1. A. V. Bitsadze, UMN, 3, 6 (28), 211 (1948).
2. M. I. Vishik, Mat. sbornik, 29, 3, 615 (1951).
3. A. V. Bitsadze, Equations of Mixed Type, 1959, p. 64.