UDC 517.946.8

ON THE DIRICHLET PROBLEM FOR CERTAIN ELLIPTIC SYSTEMS OF SECOND ORDER

E. V. Zolotareva

Consider a system of the form

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

where \(A, B, C\) are square constant matrices, and \(u=[u^1,u^2]\) is the unknown vector.

We shall assume that the characteristic determinant of system (1)

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

has no real roots, i.e., we shall consider elliptic systems of the form (1).

In this note we shall give necessary and sufficient conditions for uniqueness of the Dirichlet problem for systems of the form (1) in any simply connected domain, and we shall also give a method for constructing a solution of the Dirichlet problem in any ellipse, both for the case of weakly coupled systems and for the case of strongly coupled systems. (For the definition of weak coupling, see [1].)

It was proved in [2] that if the Dirichlet problem is unique and solvable for all possible ellipses, then it is unique and solvable in any domain. Therefore it is sufficient to clarify the questions of uniqueness for ellipses.

In this note we shall restrict ourselves to the case where the characteristic determinant (2) has distinct complex roots
\(\lambda_1=\alpha_1+i\beta_1\) and \(\lambda_2=\alpha_2+i\beta_2\), where \(\beta_1>0\), \(\beta_2>0\) (of course, \(\overline{\lambda}_1\) and \(\overline{\lambda}_2\) are also roots), since the case of multiple roots has already been considered in [3].

Thus, we shall seek a solution of system (1), regular in \(D\) and continuously differentiable up to \(\Gamma\), satisfying the boundary condition \(u|_{\Gamma}=f\), where the domain \(D\) is some ellipse, \(\Gamma\) is the boundary of the domain \(D\), and \(f(s)=[f^1(s),f^2(s)]\) is a prescribed vector-valued function of a point of the boundary. (The smoothness of \(f(s)\) will be discussed below.)

Let us note that, without loss of generality, one may consider the Dirichlet problem for the unit disk with center at the origin.

It is easy to see that a rotation of the axes and a translation of the origin are admissible without loss of generality of the problem (see, for example, [3]). Consequently, we have the right to consider the Dirichlet problem only for ellipses of the following form:

\[ \frac{x^2}{c_1^2}+\frac{y^2}{c_2^2}=1. \]

We shall now show that passage to the unit circle also does not restrict the generality of the problem. Introduce new independent variables \(\dfrac{x}{c_1}=\xi\),

\[ \frac{y}{c_2}=\eta. \]
After the change of variables we obtain a certain new system, also of the form (1),
\[ A^{*}u_{\xi\xi}+2B^{*}u_{\xi\eta}+C^{*}u_{\eta\eta}=0, \]
where
\[ A^{*}=A\frac{1}{c_1^2},\qquad B^{*}=B\frac{1}{c_1c_2},\qquad C^{*}=C\frac{1}{c_2^2}, \]
and
\[ \det\left\|A^{*}+2B^{*}\delta+C^{*}\delta^2\right\|=0 \]
will be its characteristic determinant. Denote its roots by \(\delta_1\) and \(\delta_2\). It is easy to see that
\[ \delta_j=\frac{c_2}{c_1}\lambda_j\qquad (j=1,\,2). \tag{3} \]

It is obvious that if \(\lambda_1\ne\lambda_2\), then also \(\delta_1\ne\delta_2\), and conversely.

Consequently, under stretching or compression in the direction of the coordinate axes, system (1) is transformed into some other system of the same form, and the roots of its characteristic determinant are related to the roots of the original system by relation (3).

Thus, we shall consider the Dirichlet problem only for the unit disk \(x^2+y^2=1\).

As is known [1], the general solution of system (1) has the following form:
\[ u=\operatorname{Re}\,[a\varphi_1(z_1)+b\varphi_2(z_2)], \tag{4} \]
where \(\varphi_1\) and \(\varphi_2\) are arbitrary holomorphic functions of the variables \(z_j=x+\lambda_j y\) \((j=1,2)\), respectively, and the vectors \(a=[a^1,a^2]\), \(b=[b^1,b^2]\) are solutions of the following system:
\[ \begin{cases} (A+2B\lambda_1+C\lambda_1^2)a=0,\\ (A+2B\lambda_2+C\lambda_2^2)b=0. \end{cases} \tag{5} \]

Note that the functions \(\varphi_j\) are determined from the given vector function \(u\) up to an arbitrary real constant, since
\[ u(0,0)=\operatorname{Re}\,[a\varphi_1(0)+b\varphi_2(0)]. \]

Represent \(z_j\) in the following way:
\[ z_j=x+\lambda_j y =z\,\frac{1-\lambda_j i}{2}+\bar z\,\frac{1+\lambda_j i}{2}\qquad (j=1,\,2). \tag{6} \]

The transformations (6) take the ellipses
\[ (\bar\lambda_j^{\,2}+1)z_j^2+(\lambda_j^2+1)\bar z_j^{\,2} -2(\lambda_j\bar\lambda_j+1)z_j\bar z_j =(\bar\lambda_j-\lambda_j)^2 \tag{7} \]
of the \(z_j\)-plane \((j=1,2)\) into the unit circle \(z\bar z=1\) of the \(z\)-plane, and their interiors into the interior of the circle.

Since we are considering the Dirichlet problem for the unit disk, it is therefore obvious that the functions \(\varphi_j\) must be holomorphic inside the ellipses (7) of the \(z_j\)-plane \((j=1,2)\).

Substituting expressions (6) into the general solution (4), we obtain
\[ u=\operatorname{Re}\left[ a\varphi_1\left( z\,\frac{1-\lambda_1 i}{2}+\bar z\,\frac{1+\lambda_1 i}{2} \right) +\right. \]
\[ \left. +b\varphi_2\left( z\,\frac{1-\lambda_2 i}{2}+\bar z\,\frac{1+\lambda_2 i}{2} \right) \right]. \]

Let us write out the boundary condition, denoting the points of the boundary by \(t\) \((t\bar t=1)\):

\[ f=\operatorname{Re}\left[ a\varphi_1\left(t\frac{1-\lambda_1 i}{2}+\frac{1}{t}\frac{1+\lambda_1 i}{2}\right) + b\varphi_2\left(t\frac{1-\lambda_2 i}{2}+\frac{1}{t}\frac{1+\lambda_2 i}{2}\right) \right]. \tag{8} \]

Consider the following change of variables:

\[ \omega_j=z\frac{1-\lambda_j i}{2}+\frac{1}{z}\frac{1+\lambda_j i}{2} \qquad (j=1,2). \tag{9} \]

Expression (9) can be represented in the following form:

\[ \omega_j=\gamma_j\left(z\sigma_j+\frac{1}{z\sigma_j}\right), \]

where

\[ \gamma_j=\frac{\sqrt{1+\lambda_j^2}}{2},\qquad \sigma_j=\sqrt{\frac{1-\lambda_j i}{1+\lambda_j i}}. \]

Consequently, in contrast to the transformations (6), the transformations (9) are conformal transformations consisting of two similarity transformations and one transformation carried out by the function

\[ \omega=\frac{1}{2}\left(z+\frac{1}{z}\right). \]

The ellipses defined by equations (7), under the transformations (9), are again transformed into the unit circle \(z\bar z=1\) of the \(z\)-plane, while the cuts between the foci of the ellipses are transformed into circles of radii \(r_j\), where \(r_j=\frac{1}{|\sigma_j|}<1\). Moreover, when the point \(z\) traverses the circle of radius \(r_j\), the corresponding point \(\omega_j\) traverses the indicated cut twice.

Thus, the transformations (9) conformally map the interiors of the ellipses (7) with cuts onto two annuli \((r_1<|z|<1,\ r_2<|z|<1)\) of the \(z\)-plane.

Since the functions \(\varphi_j\) are holomorphic inside the ellipses (7) and since the transformations (9) are conformal, the functions

\[ \varphi_j^*(z)=\varphi_j\left(z\frac{1-\lambda_j i}{2}+\frac{1}{z}\frac{1+\lambda_j i}{2}\right) \qquad (j=1,2) \]

expand into Laurent series in the annuli \(r_j<|z|<1\), respectively.

Introduce the notation:

\[ \begin{aligned} \varphi_1^*(z)&=\sum_{-\infty}^{\infty} a_k z^k &&\text{converges for } r_1<|z|<1,\\ \varphi_2^*(z)&=\sum_{-\infty}^{\infty} b_k z^k &&\text{converges for } r_2<|z|<1. \end{aligned} \tag{10} \]

Since to one and the same point on the cut of the ellipse (7) there correspond two symmetric points on the circle of radius \(r_j\) of the \(z\)-plane, the following equality must hold:

\[ \varphi_j^*\left(r_j e^{\,i\left(\alpha_j+\alpha_j^0\right)}\right) = \varphi_j^*\left(r_j e^{\,i\left(\alpha_j^0+\pi-\alpha_j\right)}\right) \quad (j=1,2), \tag{11} \]

where \(r_j e^{i\alpha_j}\) denotes the current points of the circle, while \(r_j e^{i\alpha_j^0}\) and \(r_j e^{i(\alpha_j^0+\pi)}\) are the points of the circle in the \(z\)-plane corresponding to the zero point on the cut of the ellipse.

Choose the larger of the radii \(r_1, r_2\). Let \(r_1>r_2\). In the annulus \(r_1<|z|<1\) both series will converge.

For \(|z|=1\), these series must satisfy the boundary condition (8), namely

\[ f=\frac12\left[ a\sum_{-\infty}^{\infty} a_k t^k + b\sum_{-\infty}^{\infty} b_k t^k + \overline a\sum_{-\infty}^{\infty} \overline{a_k}\frac1{t^k} + \overline b\sum_{-\infty}^{\infty} \overline{b_k}\frac1{t^k} \right]. \tag{12} \]

Moreover, taking into account expression (11), we shall have the following dependence between the coefficients of the series (10):

\[ a_{-k}=a_k\,\frac1{\sigma_1^{2k}},\qquad b_{-k}=b_k\,\frac1{\sigma_2^{2k}} \]

(from (12) it follows easily that \(r_j e^{i\alpha_j^0}=\dfrac{i}{\sigma_j}\)). Introduce the notation:

\[ \nu_j=\frac1{\sigma_j^2} = \frac{1+\lambda_j i}{1-\lambda_j i} \quad (j=1,2),\qquad |\nu_j|<1. \tag{13} \]

Then

\[ a_{-k}=a_k\nu_1^k,\qquad b_{-k}=b_k\nu_2^k. \tag{14} \]

Expanding the boundary vector-function \(f\) into a complex Fourier series
\[ f=\sum_{-\infty}^{\infty} f_k e^{ikt}, \]
substituting \(e^{it}\) for \(t\) in the boundary condition (12), and comparing the coefficients of \(e^{ikt}\), we obtain

\[ f_k=\frac12\left[aa_k+bb_k+\overline a\,\overline{a_{-k}}+\overline b\,\overline{b_{-k}}\right] \quad (k=0,1,2,\ldots). \tag{15} \]

Taking into account relations (14) and also writing the expression conjugate to (15), we shall have

\[ f_k=\frac12\left[aa_k+bb_k+\overline a\,\overline{a_k}\,\overline{\nu_1}^{\,k} +\overline b\,\overline{b_k}\,\overline{\nu_2}^{\,k}\right], \]

\[ f_{-k}=\frac12\left[aa_k\nu_1^k+bb_k\nu_2^k+\overline a\,\overline{a_k}+\overline b\,\overline{b_k}\right] \tag{16} \]

\[ (k=0,1,2,\ldots). \]

Since the relations obtained are vector ones (\(f_k, a, b\) are vectors), they may be regarded as a system of four equations with respect to the four unknowns: \(a_k, \overline{a_k}, b_k, \overline{b_k}\). The determinant in the unknowns of system (16) has the form

\[ \Delta_k= \left| \begin{array}{cccc} a^1 & b_1 & \overline{a^1}v_1^k & \overline{b^1}v_2^k\\ a^2 & b^2 & \overline{a^2}v_1^k & \overline{b^2}v_2^k\\ a^1v_1^k & b^1v_2^k & \overline{a^1} & \overline{b^1}\\ a^2v_1^k & b^2v_2^k & \overline{a^2} & \overline{b^2} \end{array} \right|, \qquad k=0,1,2,\ldots \]

Thus, the question of solvability of the Dirichlet problem on all possible ellipses is equivalent to the question of solvability of the linear algebraic systems (16) and to the question of convergence of the series (10).

Let us first consider the question of solvability of the systems (16), i.e., let us investigate when the determinant \(\Delta_k\) vanishes.

Note that \(\Delta_0=0\), i.e., \(a_0\) and \(b_0\) are determined up to an arbitrary real constant, which corresponds to the remark made earlier concerning the functions \(\varphi_j\) from the general solution (4).

Expanding \(\Delta_k\) \((k=1,2,\ldots)\) with respect to the first two rows and using the identity

\[ |a^1b^2-b^1a^2|^2-|a^1\overline{b^2}-\overline{b^1}a^2|^2 -(a^1\overline{a^2}-\overline{a^1}a^2)(b^1\overline{b^2}-\overline{b^1}b^2)=0, \]

we obtain

\[ \Delta_k=|a^1b^2-a^2b^1|^2(1-v_1^k\overline{v_2^k})(1-\overline{v_1^k}v_2^k) \]
\[ -|a^1\overline{b^2}-a^2\overline{b^1}|^2(v_1^k-v_2^k)(\overline{v_1^k}-\overline{v_2^k}). \tag{17} \]

In what follows we shall need the following notation:

\[ q=\left|\frac{a^1b^2-a^2b^1}{a^1\overline{b^2}-a^2\overline{b^1}}\right|, \qquad \Theta=\left|\frac{\lambda_1-\lambda_2}{\lambda_1-\overline{\lambda_2}}\right| \tag{18} \]

(it is easy to calculate that

\[ \left|\frac{\lambda_1-\lambda_2}{\lambda_1-\overline{\lambda_2}}\right| = \left|\frac{v_1-v_2}{1-v_1v_2}\right| \]).
]

Let us consider several cases.

I. Let \(a^1b^2-b^1a^2\ne0\) (i.e., we shall consider weakly coupled systems (1)) and a) let \(a^1\overline{b^2}-a^2\overline{b^1}=0\). Then \(\Delta_k\ne0\) for every \(k\).

Case I, a) is proved trivially. \(\Delta_k\ne0\) for every \(k\), since \(|v_1|<1,\ |v_2|<1\).

b) Let \(a^1\overline{b^2}-a^2\overline{b^1}\ne0\) and \(q>\Theta\). Then likewise \(\Delta_k\ne0\) for every \(k\).

Let us prove assertion I, b). Indeed, transforming equality (17), we shall have

\[ \frac{\Delta_k} {|a^1\overline{b^2}-a^2\overline{b^1}|^2(1-v_1^k\overline{v_2^k})(1-\overline{v_1^k}v_2^k)} = \]
\[ =q^2-\frac{(v_1^k-v_2^k)(\overline{v_1^k}-\overline{v_2^k})} {(1-v_1^k\overline{v_2^k})(1-\overline{v_1^k}v_2^k)}. \tag{19} \]

Our assertion will be true if we prove the following inequality:

\[ \frac{(v_1^k-v_2^k)(\overline{v_1^k}-\overline{v_2^k})} {(1-v_1^k\overline{v_2^k})(1-\overline{v_2^k}v_1^k)} \le \Theta^2. \tag{20} \]

Taking into account the notation (18), we rewrite inequality (20) in the following form:

\[ 1-\frac{(1-v_1^k\overline{v_1^k})(1-v_2^k\overline{v_2^k})} {(1-v_1^k\overline{v_2^k})(1-v_2^k\overline{v_1^k})} \le 1-\frac{(1-v_1\overline{v_1})(1-v_2\overline{v_2})} {(1-v_1\overline{v_2})(1-\overline{v_1}v_2)}. \]

Consequently, we must prove the following inequality:

\[ \frac{(1-\nu_1^k\overline{\nu}_1^k)(1-\nu_2^k\overline{\nu}_2^k)} {(1-\nu_1^k\overline{\nu}_2^k)(1-\overline{\nu}_1^k\nu_2^k)} \geq \frac{(1-\nu_1\overline{\nu}_1)(1-\nu_2\overline{\nu}_2)} {(1-\nu_1\overline{\nu}_2)(1-\overline{\nu}_1\nu_2)} . \]

Dividing by

\[ \frac{(1-\nu_1\overline{\nu}_1)(1-\nu_2\overline{\nu}_2)} {(1-\nu_1\overline{\nu}_2)(1-\overline{\nu}_1\nu_2)} > 0, \]

we shall have

\[ \frac{(1+\nu_1\overline{\nu}_1+\ldots+\nu_1^{k-1}\overline{\nu}_1^{\,k-1}) (1+\nu_2\overline{\nu}_2+\ldots+\nu_2^{k-1}\overline{\nu}_2^{\,k-1})} {(1+\nu_1\overline{\nu}_2+\ldots+\nu_1^{k-1}\overline{\nu}_2^{\,k-1}) (1+\overline{\nu}_1\nu_2+\ldots+\overline{\nu}_1^{\,k-1}\nu_2^{k-1})} \geq 1. \]

The inequality will only be strengthened if we transform it in the following way:

\[ \frac{[1+|\nu_1|^2+\ldots+|\nu_1|^{2(k-1)}] [1+|\nu_2|^2+\ldots+|\nu_2|^{2(k-1)}]} {[1+|\nu_1||\nu_2|+\ldots+|\nu_1|^{k-1}|\nu_2|^{k-1}]^2} \geq 1. \]

But this is the well-known Cauchy inequality, i.e., inequality (20) is valid.

c) Let \(a^1\overline{b}^2-a^2\overline{b}^1\ne 0\) and \(q\leq \Theta\). Then \(\Delta_k\) can vanish only a finite number of times. Moreover, there always exists an ellipse for which at least one \(\Delta_{k_0}=0\).

Indeed, from inequality (17) it is easy to see that

\[ \lim_{k\to\infty}\Delta_k=|a^1b^2-b^1a^2| \]

(but \(|a^1b^2-b^1a^2|\ne 0\) by assumption). In other words, there is an integer \(N\) such that for all \(k>N\) all \(\Delta_k\ne 0\), i.e., \(\Delta_k\) can vanish only a finite number of times.

We now turn to the proof of the second half of our assertion. The magnitude of the determinant \(\Delta_k\) for fixed \(k\) depends on the ellipse on which the Dirichlet problem is considered. Indeed, as follows from relation (3), the roots of the characteristic polynomial depend on the ratio of the semiaxes of the ellipse. At the same time, the quantity \(q\) does not depend on the ratio of the semiaxes of the ellipse, since the vectors \(a\) and \(b\) do not change under transformation (3).

Introduce the following notation:

\[ F_k(x)= \frac{[\nu_1^k(x)-\nu_2^k(x)][\overline{\nu}_1^k(x)-\overline{\nu}_2^k(x)]} {[1-\nu_1^k(x)\overline{\nu}_2^k(x)][1-\overline{\nu}_1^k(x)\nu_2^k(x)]}, \]

where

\[ \nu_j(x)=\frac{1+ix\lambda_j}{1-ix\lambda_j} \quad\text{and}\quad x=\frac{c_1}{c_2} \quad \text{(see (3) and (13)).} \]

Then expression (19), written for the ellipse with semiaxes \(c_1\) and \(c_2\) \(\left(x=\dfrac{c_1}{c_2}\right)\), will have the form

\[ \frac{\Delta_k(x)} {|a^1b_2-a^2b_1|^2(1-\nu_1^k(x)\overline{\nu}_2^k(x))(1-\overline{\nu}_1^k(x)\nu_2^k(x))} = q^2-F_k(x). \]

Our assertion will be proved if we show the existence of an \(x_0\) such that \(F_k(x_0)=q^2\) for at least one \(k\). Represent the function \(F_k(x)\) in the following way:

\[ F_k(x)= \frac{[\nu_1(x)-\nu_2(x)][\overline{\nu}_1(x)-\overline{\nu}_2(x)] [\nu_1^{k-1}(x)+\nu_1^{k-2}(x)\nu_2(x)+\ldots+\nu_2^{k-1}(x)]} {[1-\nu_1(x)\overline{\nu}_2(x)][1-\overline{\nu}_1(x)\nu_2(x)] [1+\nu_1(x)\overline{\nu}_2(x)+\ldots+\nu_1^{k-1}(x)\overline{\nu}_2^{\,k-1}(x)]} \times \]

\[ \times \frac{[\overline{v}_1^{\,k-1}(x)+\overline{v}_1^{\,k-2}(x)\overline{v}_2(x)+\ldots+\overline{v}_2^{\,k-1}(x)]} {[1+\overline{v}_1(x)v_2(x)+\ldots+\overline{v}_1^{\,k-1}(x)v_2^{\,k-1}(x)]}. \]

It is easy to calculate that

\[ \frac{[v_1(x)-v_2(x)][\overline{v}_1(x)-\overline{v}_2(x)]} {[1-v_1(x)\overline{v}_2(x)][1-\overline{v}_1(x)v_2(x)]} =\Theta^2, \tag{21} \]

i.e., the expression (21) does not depend on \(x\).

Hence we have that \(\lim_{x\to 0} F_k(x)=\Theta^2\). Since, by assumption, \(\Theta>q\), it follows that \(F_k(0)>q^2\) for all \(k\).

On the other hand, \(\lim_{k\to\infty}F_k(1)=0\). Consequently, there exists a number \(k_0\) such that \(F_{k_0}(1)<q^2\).

Since \(F_{k_0}(x)\) is a continuous positive function, there will always be found such a \(0<x_0<1\) that \(F_{k_0}(x_0)=q^2\).

That is, in other words, there will always be found an ellipse \(\left(x_0=\dfrac{c_1}{c_2}\right)\) for which \(\Delta_{k_0}(x_0)=0\).

II. Let \(a^1b^2-a^2b^1=0\) (i.e., we shall consider strongly coupled systems (1)) and a) let \(a^1\overline{b}^{\,2}-\overline{b}^{\,1}a^2\ne0\), and let \(\dfrac{v_1}{v_2}\) not be any root of unity; then \(\Delta_k\ne0\) for any \(k\).

If \(\dfrac{v_1}{v_2}\) is not a root of unity, then \(v_1^k\ne v_2^k\) for any \(k\), i.e., \(\Delta_k\ne0\) always.

b) Let \(a^1\overline{b}^{\,2}-\overline{b}^{\,1}a^2\ne0\) and \(\left(\dfrac{v_1}{v_2}\right)^{k_0}=1\). Then \(\Delta_k\) vanishes in infinitely many cases.

Indeed, if \(v_1^{k_0}=v_2^{k_0}\), then also \(v_1^{pk_0}=v_2^{pk_0}\), where \(p\) is any positive integer.

Let us also note that the case when \(a^1b^2-a^2b^1=0\) and \(a^1\overline{b}^{\,2}-a^2\overline{b}^{\,1}=0\) cannot occur by virtue of the ellipticity of system (1).

Thus, we have investigated the question of the solvability of systems (16). We now pass to the question of the convergence of the series (10). Suppose that we have found the coefficients \(a_k\) and \(b_k\) from systems (16); let us determine whether the series obtained by us will converge.

From systems (16) we shall have

\[ a_k= \frac{f_k^1A_{11}+f_k^2A_{21}+f_{-k}^1A_{31}+f_{-k}^2A_{41}} {\Delta_k}, \]

\[ b_k= \frac{f_k^1A_{12}+f_k^2A_{22}+f_{-k}^1A_{32}+f_{-k}^2A_{42}} {\Delta_k}, \tag{22} \]

where \(A_{ik}\) \((i=1,2,3,4;\ k=1,2)\) are the cofactors of the elements of the first and second columns.

Let us also recall that \(\lim_{k\to\infty}\Delta_k=|a^1b^2-a^2b^1|^2\). Consider two cases:

I. Let \(a^1b^2-a^2b^1\ne0\) (i.e., system (1) is weakly coupled). In this case the denominators of the expressions (22) do not tend to zero as \(k\to\infty\).

If we require that the boundary vector-function have a continuous first derivative, then its Fourier coefficients will satisfy the following inequalities:

\[ |f_k|<\frac{c}{k^2}\qquad (k=\pm 1,\ \pm 2,\ \pm 3,\ldots). \]

Consequently,

\[ |a_k|<\frac{M}{k^2},\qquad |b_k|<\frac{N}{k^2}\qquad (k=1,2,3,\ldots). \tag{23} \]

Taking into account relations (14) and estimates (23), we obtain that

\[ \varphi_1^*(z)=\sum_{-\infty}^{\infty} a_k z^k \quad \text{converges for } |\nu_1|\le |z|\le 1, \]

\[ \varphi_2^*(z)=\sum_{-\infty}^{\infty} b_k z^k \quad \text{converges for } |\nu_2|\le |z|\le 1, \]

i.e., the series (10) converge even in a wider domain than we required \((r_j=\sqrt{|\nu_j|})\).

Since the series (10) converge in the indicated annuli, the series

\[ \varphi_1(\omega_1)=\varphi_1^*(z(\omega_1)) =\sum_{-\infty}^{\infty}\frac{a_k}{2^k} \left(\omega_1+\sqrt{\omega_1^2+1+\lambda_1^2}\right)^k, \]

\[ \varphi_2(\omega_2)=\varphi_2^*(z(\omega_2)) =\sum_{-\infty}^{\infty}\frac{b_k}{2^k} \left(\omega_2+\sqrt{\omega_2^2+1+\lambda_2^2}\right)^k \]

also converge, i.e., the functions \(\varphi_j(\omega_j)\) \((j=1,2)\) are holomorphic inside the ellipses (7) with cuts.

By virtue of the equality (11), the functions \(\varphi_j(z_j)\) are holomorphic already inside the solid ellipses (7), i.e., the solution found,

\[ u=\operatorname{Re}\left[ a\sum_{-\infty}^{\infty}\frac{a_k}{2^k} \left(z_2+\sqrt{z_1^2+1+\lambda_1^2}\right)^k + b\sum_{-\infty}^{\infty}\frac{b_k}{2^k} \left(z_2+\sqrt{z_2^2+1+\lambda_2^2}\right)^k \right] \tag{24} \]

is suitable.

II. Let \(a^1b^2-a^2b^1=0\) (i.e., system (1) is strongly coupled). In this case the denominators of expressions (22) tend to zero as \(k\to\infty\). Writing expressions (22) in greater detail and taking into account that, by assumption, \(a^1b^2=b^1a^2\), we shall have

\[ a_k=M_1\frac{f_k^1\nu_2^k}{\nu_1^k-\nu_2^k} +M_2\frac{f_k^2\nu_2^k}{\nu_1^k-\nu_2^k} +M_3\frac{f_{-k}^1}{\nu_1^k-\nu_2^k} +M_4\frac{f_{-k}^2}{\nu_1^k-\nu_2^k}, \tag{25} \]

\[ b_k=N_1\frac{f_k^1\nu_1^k}{\nu_1^k-\nu_2^k} +N_2\frac{f_k^2\nu_1^k}{\nu_1^k-\nu_2^k} +N_3\frac{f_{-k}^1}{\nu_1^k-\nu_2^k} +N_4\frac{f_{-k}^2}{\nu_1^k-\nu_2^k}, \]

where \(M_e, N_e\) are constants independent of \(k\) \((e=1,2,3,4)\).

Obviously, if \(\dfrac{C_2}{k^p}<|f_k|<\dfrac{C_1}{k^p}\) (\(p\) is any natural number), then the series (10) will diverge. That is, whatever smoothness we require of \(f\), this cannot ensure the convergence of the series (10).

If we require that the boundary vector-function satisfy the following estimates:

\[ |f_k^1a^2+f_k^2a^1|<C|v_2|^k \qquad (k=\pm1,\ \pm2,\ \pm3,\ldots) \]

(in other words, if we require that the function \(f^*(t)=f^1(t)a^2+f^2(t)a^1\) can be continued inside the circle up to the circumference of radius \(|v_2|\)), then the series (10) will converge.

Indeed, it is easy to compute that

\[ M_3=\bar a^2(\bar b^1b^2-\bar b^2b^1),\qquad M_4=\bar a^1(\bar b^1b^2-\bar b^2b^1), \]

\[ N_3=\bar b^2(\bar a^1a^2-\bar a^2a^1),\qquad N_4=\bar b^1(\bar a^1a^2-\bar a^2a^1). \]

Taking these expressions into account, from the relations (25) we obtain

\[ |a_k|\leq \frac{M|v_2|^k}{|v_1|^k-|v_2|^k}; \qquad |b_k|\leq \frac{N|v_2|^k}{|v_1|^k-|v_2|^k} \qquad (k=1,2,\ldots). \tag{26} \]

Consequently, the series

\[ \varphi_1^*(z)=\sum_{-8}^{\infty}a_kz^k,\qquad \varphi_2^*(z)=\sum_{-\infty}^{\infty}b_kz^k, \]

converge for

\[ \frac{|v_2|^2\bigl(|v_1|^k-|v_2|^k\bigr)} {|v_1|^{k+1}-|v_2|^{k+1}} < z < \frac{|v_1|^{k+1}-|v_2|^{k+1}} {|v_2|\bigl(|v_1|^k-|v_2|^k\bigr)}, \]

and since the number

\[ \frac{|v_1|^{k+1}-|v_2|^{k+1}} {|v_2|\bigl(|v_1|^k-|v_2|^k\bigr)} >1, \]

the series (10) converge for \(|v_2|\leq |z|\leq 1\).

Summarizing all that has been said above, one may draw the following conclusion. If the characteristic determinant (2) of the system (1) has distinct roots \(\lambda_1\) and \(\lambda_2\) (respectively \(\bar\lambda_1\) and \(\bar\lambda_2\)) and the system (1) is weakly coupled, then

1) for \(q>\Theta\) the Dirichlet problem for any domain is always solvable and unique. Its solution is given by formula (24);

2) for \(q\leq\Theta\) the homogeneous Dirichlet problem may have a finite number of nontrivial linearly independent solutions; for solvability of the nonhomogeneous problem the same number of orthogonality conditions must be imposed on the boundary vector-function. Moreover, there is always an ellipse for which uniqueness of the Dirichlet problem fails.

Let us note that \(\Theta=\left|\dfrac{\lambda_1-\lambda_2}{\lambda_1-\bar\lambda_2}\right|<1\); consequently, for \(q>1\) the Dirichlet problem is solvable and unique.

If, however, the system (1) is strongly coupled, then

1) when \(\dfrac{v_1}{v_2}\ne\varepsilon_k\) (\(\varepsilon_k\) is some root of unity), the homogeneous Dirichlet problem has only the trivial solution; for solvability of the nonhomogeneous problem it is necessary to require that the function \(f^*(t)=f^1(t)a^2+\)

\(+ f^{2}(t)a^{1}\) can be continued inside the unit disk up to the circle of radius \(|v_{2}|\). The solution is again given by formula (24).

2) For \(\dfrac{v_{1}}{v_{2}}=\varepsilon_{k}\), the homogeneous Dirichlet problem has infinitely many linearly independent nontrivial solutions; for the solvability of the nonhomogeneous problem, in addition to the requirement that the function \(f^{*}(t)\) be continuable, one must impose infinitely many orthogonality conditions.

References

1. Bitsadze A. V. An equation of mixed type. Publishing House of the Academy of Sciences of the USSR, Moscow, 1959, pp. 58–66.

2. Din Sya-si, Chzhan Tun, Ma Zhu-nyan, Van Kon-tin. Scince Record, 4, 3, 1960, 160.

3. Zolotareva E. V. On the uniqueness of the Dirichlet problem for certain elliptic systems. Siberian Mathematical Journal, No. 2, 1967.

Received by the editors
March 9, 1966.

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