Exterior Boundary-Value Poincaré Problem for Elliptic Systems of the Second Order with Two Independent Variables

Abstract

This paper offers a number of examples showing that in the case of two independent variables the uniform ellipticity of a linear system of differential equations with partial derivatives of the second order, which fulfills condition (3), do not always cause the normal solvability of formulated exterior elliptic problems in the sense of Noether.

Nevertheless, from the system of differential equations with partial derivatives of elliptic type it is possible to choose, under certain additional conditions, classes which are normally solvable in the sense of Noether.

This paper also shows that for the so-called decomposed system of differential equations, with partial derivatives of an elliptic type in the case of exterior regions, the Noether theorems are valid.

1 Introduction

Among the linear elliptic boundary-value problems, Poincare problem is of great importance, side by side Dirichlet and Neumann problems. Numerous processes occurring in the continuum (for example, sea tides, see [1], as well as [2]) can be simulated in terms of this problem. This problem differs essentially from Dirichlet and Neumann problem in the fact that it is normally solvable according to Noether under rather general assumptions (i.e. Noether theorems known in the theory of singular integral equations are valid for it, see [1, 3–10]).

In the process of the investigation of Dirichlet and Neumann problems for external regions, certain difficulties arose due to the complex, behavior of the solutions of elliptic equations at infinity. This fact was revealed on the example of Helmholtz equation simulating wave processes in the linear formulation. The irradiation principle is of great importance for the theory of wave processes investigated in physics, technology, ecology and natural science. The existence, uniqueness and stability of the solutions of the mentioned problems for Helmholtz equation in infinite regions with boundary components representing closed Lyapunov’s surfaces (in three-dimensional case) and closed Lyapunov’s curves (in two-dimensional case) were established in the class of functions satisfying the condition of Sommerfeld’s irradiation at infinity [11–13].

2 Statements and Definitions

Let $S$ be a closed Lyapunov curve on the plane of complex variable $z= x + i y$, and $D^{-}$ a simply connected domain with boundary $S$, containing an infinitely distant point of the plane (exterior domain).

This paper is devoted to the study of the exterior boundary-value Poincaré problem for an elliptic system of the type

$$ A \frac{\partial ^{2} u}{\partial x^{2}} + 2 B \frac{\partial ^{2} u}{\partial x \partial y} + C \frac{\partial ^{2} u}{\partial y^{2}} = 0, $$

(1)

where $A = (a_{jk})$, $B = (b_{jk})$, $C = (c_{jk})$ are given real $n \times n$ constant matrices and $w = (u_{1}, \ldots , u_{n})$ the search real vector.

Definition 1

Vector $w(x,y)$, having continuous derivatives of the second order bounded at infinity and satisfying equation (1) in domain $D^{-}$, will be called the regular solution of this equation.

Let real $n \times n$ matrices $P=(p_{jk})$, $Q=(q_{jk})$, $R=(r_{jk})$ and the real vector $f = (f_{1},\ldots ,f_{n})$ be given on $S$. Since $S$ is a rectifiable curve, it is assumed that $P$, $Q$, $R$ and $f$, are given as the functions of arc length $s$ of the curve reckoned from the fixed point on $S$ in the direction that domain $D^{-}$ leaves on the right.

Definition 2

The exterior boundary-value Poincaré problem is assumed as the problem of determination of regular solution $w(z) = w(x,y)$ in domain $D^{-}$ of equation (1), satisfying the boundary-value condition on $S$

$$ P \frac{\partial u}{\partial x} + Q \frac{\partial u}{\partial y} + R w = f. $$

(2)

It is assumed that the finite value of vector $w(z)$ and its first derivatives inside $D^{-}$ on $S$ exist. Note that when $n=1$ and $R\equiv 0$, problem (1), (2) is called an oblique derivative problem. Since in each one of the $n$ boundary-value conditions (2) there are oblique derivatives along different directions for various components of search bounded vector $w = (u_{1},\ldots ,u_{n})$ in $D^{-}$, it was necessary to call problem (2) that of Poincaré. At $p_{jk} = q_{jk} = 0$, $j \neq k$, $p_{jj} = \cos \widehat{\nu x}$, $q_{jj} = \cos \widehat{\nu y}$, $j,k = 1, \ldots ,n$, where $\nu$ inner normal to $S$ at the point $(x,y)$ relative to $D^{-}$, boundary-value condition (2) is somewhat different to that of Neumann. If $P \equiv Q \equiv 0$ and $\det (R) \neq 0$ on $S$, condition (2) is transferred into the boundary-value condition of the Dirichlet problem.

The case in which the roots of the characteristic determinant of system (1) are complex will be considered throughout this paper, such system is called elliptic [14]. At $n=1$ equality (1) represents an elliptic equation of the second order, which is reduced to Laplace’s equation by means of non-singular (non-degenerate) affine transformation of $x$, $y$ variables.

3 Examples

The domain $D^{-}$ is defined as the exterior of the unit circle $|z| \le 1$ for the examples only in this section.

For Laplace’s equation, assuming that $S$ is Lyapunov curve, $P$, $Q$, $R$, $f$ are Hölder continuous [15] on the whole $S$ i.e. $p, q, r, f \in C^{0,h}(S)$, and

$$ \det (P + i Q) \neq 0, $$

(3)

for problem (2) the normal solvability according to Noether holds [14, 16], as well as the uniqueness of the solution to Dirichlet problem.

In the case of the elliptic system of type (1), the investigation of exterior problems of Dirichlet, Neumann and Poincaré becomes rather difficult. Actually, at $n > 1$ as in the case of bounded domains [16], the requirement of uniform ellipticity of system (1) does not always guarantee the normal solvability (accordingly to Fredholm and Noether) of exterior problem (2). This is easy to check by the example of Bidsatze elliptic system¹

$$ \left \{ \begin{aligned} \frac{\partial ^{2} u_{1}}{\partial x^{2}} - 2 \frac{\partial ^{2} u_{2}}{\partial x \partial y} - \frac{\partial ^{2} u_{1}}{\partial y^{2}} & = 0, \\ \frac{\partial ^{2} u_{2}}{\partial x^{2}} + 2 \frac{\partial ^{2} u_{1}}{\partial x \partial y} - \frac{\partial ^{2} u_{2}}{\partial y^{2}} & = 0, \end{aligned} \right . $$

(4)

when domain $D^{-}$ is the exterior of unit circle $|z| \le 1$.

This system can be written in the following form:

$$ \frac{\partial ^{2} w}{\partial \bar{z}^{2}} = 0 $$

(5)

where

$$\begin{gathered} 2\dfrac{\partial \omega}{\partial \bar{z}} = \dfrac{\partial \omega}{\partial x} + i \dfrac{\partial \omega}{\partial y} \\ w = u_{1}(x,y) + i u_{2}(x,y) \end{gathered}$$

From (5) one can conclude immediately that the regular solution to system (4) can be given in the following way:

$$w = u_{1}(z) + i u_{2}(z) = u_{1}(x,y) + i\, u_{2}(x,y) = \bar{z} \varphi (z) + \psi (z) $$

for any arbitrary analytic function $\varphi (z)$ and $\psi (z)$ of variable $z$ in the region $D^{-}$ [16].

Homogeneous Dirichlet Problem

It is easy to check that homogeneous Dirichlet problem

$$ \begin{gathered} u_{1}(t) = u_{2}(t) = 0, \quad t \in S, \qquad P = Q = \begin{pmatrix} 0 & 0 \\ 0 & 0 \end{pmatrix} , \quad R = \begin{pmatrix} 1 & 0 \\ 0 & 1 \end{pmatrix} \\ \det (P + i Q) = 0, \qquad \det (R) = 1, \qquad f(t) = (f_{1}(t),f_{2}(t)), \qquad x + i y = t = e^{i\theta} \end{gathered} $$

for system (4), in region $D^{-}$, has an infinite set of linearly independent regular solutions,

$$ \omega _{k}(z) = u_{1k}(z) + i u_{2k}(z) = \bar{z} z^{-k} - z^{-(k+1)}, \qquad k \ge 1,\quad z\in D^{-} $$

bounded at infinity.

Homogeneous Neumann Problem

On the other hand, let us consider the following homogeneous Neumann problem in the above region $D^{-}$

$$ \begin{gathered} \left . \begin{aligned} \frac{\partial u_{1}}{\partial \nu} &= 0 \\ \frac{\partial u_{2}}{\partial \nu} &= 0 \end{aligned} \right \}\qquad P = \begin{pmatrix} \cos \widehat{\nu x} & 0 \\ 0 & \cos \widehat{\nu x} \end{pmatrix} , \qquad Q = \begin{pmatrix} \cos \widehat{\nu y} & 0 \\ 0 & \cos \widehat{\nu y} \end{pmatrix} , \\ R = \begin{pmatrix} 0 & 0 \\ 0 & 0 \end{pmatrix} , \qquad f(t)=(f_{1}(t),f_{2}(t)) \qquad t = e^{i\theta}. \end{gathered} $$

In this case, $\cos \nu y = \sin \nu x$

$$\begin{array}{c}P+iQ=\left(\begin{array}{cc}cos\widehat{\nu x}& 0\\ 0& cos\widehat{\nu x}\end{array}\right)+i\left(\begin{array}{cc}sin\widehat{\nu x}& 0\\ 0& sin\widehat{\nu x}\end{array}\right)\hfill \\ \hfill =\left(\begin{array}{cc}cos\widehat{\nu x}+isin\widehat{\nu x}& 0\\ 0& cos\widehat{\nu x}+isin\widehat{\nu x}\end{array}\right)\end{array}$$

$$\begin{array}{c}det(P+iQ)={(cos\widehat{\nu x}+isin\widehat{\nu x})}^2={cos}^2\widehat{\nu x}-{sin}^2\widehat{\nu x}+2icos\widehat{\nu x}sin\widehat{\nu x}\hfill \\ \hfill =cos2\widehat{\nu x}+isin2\widehat{\nu x}=exp(i2\widehat{\nu x})\ne 0\end{array}$$

where $\theta = 2\widehat{\nu x}$.

This problem, as linearly independent regular solutions, has functions

$$ \begin{aligned} \omega _{0} &= 1, \\ \omega _{k}(z) &= u_{1k}(z) + i\, u_{2k}(z) = \bar{z} \frac {1}{z^{k}} - \frac{k-1}{k+1} \frac {1}{z^{k+1}}, \qquad k=1,\ldots . \end{aligned} $$

The above example shows that the fulfillment of condition (3) does not always guarantee the normal solvability of problem (4), (2) in the sense of Noether.

Neumann Problem

One should not assume, however, that for system (4) there is no problem of type (2) with the property of normal solvability. To this end, let us assume that in boundary-value condition (2) matrices $P$, $Q$, $R$ are selected as follows

$$ \begin{gathered} P = \begin{pmatrix} 0 & 0 \\ 1 & 0 \end{pmatrix} , \qquad Q = \begin{pmatrix} 0 & 0 \\ 0 & -1 \end{pmatrix} ,\qquad R = \begin{pmatrix} 1 & 0 \\ 0 & 0 \end{pmatrix} , \\ \det (P + i Q) = \det \begin{pmatrix} 0&0 \\ 1&-i\end{pmatrix} =0,\qquad f(t) = (f_{1}(t),f_{2}(t))\qquad t = e^{i\theta}. \end{gathered} $$

In the considered case the search regular solution will be as follows,

$$\begin{array}{c}\omega (z)=-\frac{\overline{z}}{4\pi i}{\int }_0^{2\pi }\frac{t+z}{t(t-z)}{f}_2(t)dt+\frac{1}{4\pi i}{\int }_0^{2\pi }\frac{t+z}{zt(t-z)}{f}_2(t)dt\hfill \\ \hfill -\frac{1}{\pi i}{\int }_0^{2\pi }\frac{f_1(t)}{t-z}dt+\frac{1}{2\pi i}{\int }_0^{2\pi }\frac{f_1(t)}{t}dt+iKz\in D^{-}\end{array}$$

$$ w(z) = u_{1}(z)+ u_{2}(z) = \bar{z} \varphi (z) + \psi (z), \qquad z \in D^{-}, t\in S. $$

(6)

This solution fulfills (4) and boundary-value condition (2) when $\int _{0}^{2\pi} f_{2}(\theta )\, d\theta = 0$, $t=e^{i\theta}$.

Unlike the Neumann problem considered above, condition (3) might be fulfilled and problem (4), (2) is normally solvable. To confirm this assumption we can consider problem (4), (2) in the assumption that

$$ \begin{gathered} P = \begin{pmatrix} 1&0 \\ 0&1\end{pmatrix} ,\quad Q = \begin{pmatrix} 0&-1 \\ -1&0\end{pmatrix} ,\quad R = \begin{pmatrix} 0&0 \\ 0&0\end{pmatrix} , \\ \det (P+iQ) = \det \begin{pmatrix} 1&-i \\ -i &1\end{pmatrix} = 2,\qquad f(t)= (f_{1}(t),f_{2}(t)), \qquad t=e^{i\theta}. \end{gathered} $$

(7)

On the basis of (6) and (7), let us write the boundary-value condition (2) in the form

$$ 2 \Re \frac{\partial \omega}{\partial \bar{t}} = 2 \Re \varphi (t) = f_{1}(t), \qquad t\in S, $$

(8)

$$ 2 \Im \frac{\partial \omega}{\partial t} = 2 \Im \left [\bar{t} \varphi '(t) + \psi '(t)\right ] = f_{2}(t), \quad t\in S. $$

(9)

Taking into the account (6) one can write

$$ \varphi (z) = -\frac{1}{4\pi i} \int _{0}^{2\pi} \frac{t+z}{t(t-z)} f_{2}(t) \, dt. $$

(10)

As the point $z=x+iy$ from the region $D^{-}$ approaches the boundary point $t\in S$, on the basis of considering the class of analytical functions in the domain $D^{-}$ that vanish at infinity, from (8) and by the Morera and Cauchy theorems, one has

$$ \varphi (t) = 0 = \int _{0}^{2\pi} f_{1}(t)\, dt \Leftrightarrow f_{1}(t) \text{ is analytical.} $$

Therefore, in the class of analytical functions in the domain $D^{-}$ vanishing at infinity, problem (8) is solvable if and only if

$$ \int _{0}^{2\pi} f_{1}(t)\, dt = 0, \qquad t=e^{i\theta}. $$

(11)

Its solution is unique, being given by

$$ \varphi (z) = -\frac{1}{4\pi i} \int _{0}^{2\pi} \frac{t+z}{t(t-z)} f_{1}(t) \, dt. $$

On the other hand, in the case of (9) we obtain

$$ \int _{0}^{2\pi}f_{2}(\theta )\, d\theta = 0, $$

(12)

and when condition (12) is fulfilled, the analytical function $\psi (z)$ in $D^{-}$, vanishing at infinity, can be defined by formula

$$ \psi '(z) = - \frac{\varphi '(z)}{z} - \frac {1}{4\pi} \int _{0}^{2 \pi}\frac{t+z}{t(t-z)}f_{2}(t)\,dt. $$

(13)

Function $\psi (z)$ can be determined from (13) as a result of integration up to the arbitrary complex constant. Therefore, it is possible to conclude that for the solvability of non-homogeneous problem (4), (2), when matrices $P$, $Q$, $R$ are given by formulas (7), it is necessary and sufficient that functions $f_{1}$ and $f_{2}$ in the right parts of the boundary-value condition (2) satisfy equalities (11) and (12). When these equalities are fulfilled the solution of the problem can be defined up to the arbitrary additive constant.

Conclusion

The above examples show the importance of the definition of the classes of elliptic systems for which the Poincaré problem, in the case of exterior domains, is normally solvable according to Noether.

4 Decomposable System

Consider the linear system of equations in partial derivatives of type (1) in the case when the characteristic determinant

$$ \det (A \lambda ^{2} + 2 B \lambda + C) \neq 0 $$

for any $\lambda \in \mathbb{R}$. It is known that such systems are called elliptic [14]. For reference, the sixth chapter of [18] is devoted entirely to de Poincare problem for second order linear elliptic systems.

Definition 3

System (1) is called decomposable [14], if $A$, $B$, $C$ are diagonal matrices, i.e. $A = (a_{jk})$, $B = (b_{jk})$, $C = (c_{jk})$,

$$ a_{jk} = \textstyle\begin{cases} a_{j}, & k = j, \\ 0, & k\neq j, \end{cases}\displaystyle \quad b_{jk} = \textstyle\begin{cases} b_{j}, & k = j, \\ 0, & k\neq j, \end{cases}\displaystyle \quad c_{jk} = \textstyle\begin{cases} c_{j}, & k = j, \\ 0, & k\neq j, \end{cases}\displaystyle \quad j,k = 1,\ldots ,n. $$

Let $A$, $B$, $C$ be constant diagonal matrices, meaning that the considered system is decomposed into $n$ equations.

$$ a_{j} \frac{\partial ^{2} u_{j}}{\partial x^{2}} + 2 b_{j} \frac{\partial ^{2} u_{j}}{\partial x \partial y} + c_{j} \frac{\partial ^{2} u_{j}}{\partial y^{2}} = 0, \qquad j = 1, \ldots , n. $$

(14)

In these assumptions the ellipticity of system (1) means that all quadratic forms

$$ Q_{j}(\lambda _{1}, \lambda _{2}) = a_{j} \lambda _{1}^{2} + 2 b_{j} \lambda _{1} \lambda _{2} + c_{j} \lambda _{2}^{2}, \qquad j = 1, \ldots ,n, $$

are positively definite.

Definition 4

The regular solution of equation (14) at each $j=1,\ldots ,n$ in $D^{-}$ will be called the bounded in $D^{-}$ twice-continuously differentiable function $u_{j}(x,y)$, satisfying this equation in each finite point in $D^{-}$.

Likewise, it is possible to verify directly that function $u_{j}(x,y)$ defined by formula

$$ u_{j}(x,y) = \Re \varphi _{j}(z_{j}) $$

(15)

where $\varphi _{j}(z_{j})$ is the arbitrary bounded analytical function in $D^{-}$ domain of complex variable

$$ \begin{aligned} z_{j} &= \frac {1}{\sqrt{a_{j}}} x + i \left ( \frac{\sqrt{a_{j}}}{\delta _{j}} y - \frac{b_{j}}{\delta _{j}\sqrt{a_{j}}}x\right ), \\ \delta _{j}^{2} &= a_{j} c_{j} - b_{j}^{2}, \end{aligned} $$

gives regular solutions of equation (14) in $D^{-}$. The proof of this proposition is available in the Appendix of this paper.

It is evident that solution $u_{j}(x,y)$ of equation (14), regular in $D^{-}$, function $\varphi _{j}(z_{j})$ can be defined up to an arbitrary imaginary constant. Without loss of generality, it can be assumed that function $\varphi _{j}(z_{j})$ in formula (15) satisfies condition

$$\Im \varphi _{j}(\infty ) = 0. $$

The following theorem will be taking into account below:

Theorem 1

Representation theorem, Vekua [19]

For each bounded analytical function $\varphi _{j}(z_{j})$ of $C^{1,h}(D^{-} \cup S)$ above considered²there exists unique real function $\mu _{j}(s)$ of the class $C^{0,h} (S)$ so that

$$ \varphi _{j}(z_{j}) = \int _{S}\ln \left (1-\frac{z_{j}}{\tau}\right ) \mu _{j}(\tau )\, d s_{\tau}. $$

(16)

Below, the Poincaré problem for decomposable system (14) is investigated and formulated as follows:

Theorem 2

Poincare problem for decomposable systems

Let’s consider the decomposable system with boundary-value conditions

$$ \begin{gathered} a_{j} \frac{\partial ^{2} u_{j}}{\partial x^{2}} + 2 b_{j} \frac{\partial ^{2} u_{j}}{\partial x \partial y} + c_{j} \frac{\partial ^{2} u_{j}}{\partial y^{2}} = 0, \qquad j = 1, \ldots , n, \\ P \frac{\partial u}{\partial x} + Q \frac{\partial u}{\partial y} = f(t), \qquad x + i y = t \in S, \end{gathered} $$

(17)

where $a_{j}$, $b_{j}$, $c_{j}$ are constant values, $j=1,\ldots ,n$, and $P$, $Q$, $R$ given real $n\times$ matrices on $S$ satisfying Hölder conditions, $f(t)=(f_{1}(t),\ldots ,f_{n}(t))$ given real vector on $S$ satisfying Hölder’s condition.

The solution to this problem is equivalent to finding the solution to a system of singular integral equation.

Proof

The proof is based on Theorem 1.

Returning to formula (15), we can write boundary-value condition (17) as

$$\begin{array}{c}\mathrm{\Re }\sum _{j=1}^n[p_{jk}(\frac{1}{\sqrt{a_j}}-i\frac{b_j}{{\delta }_j\sqrt{a_j}})+i{q}_{jk}\frac{\sqrt{a_j}}{{\delta }_j}]{\phi }_j^{\prime }(t)=f_k(t),\hfill \\ \hfill k=1,\dots ,n,t(s)\in S.\end{array}$$

(18)

Taking symbols $\alpha = (\alpha _{jk})$, $\beta =(\beta _{jk})$,

$$\begin{aligned} \alpha _{jk}(t) &= \Re \left [ p_{jk}(t) \left ( \frac {1}{\sqrt{a_{j}}} - i \frac{b_{j}}{\delta _{j}\sqrt{a_{j}}} \right ) + i q_{jk}(t) \frac{\sqrt{a_{j}}}{\delta _{j}} \right ] \pi i \bar{t}', \end{aligned}$$

(19)

$$\begin{aligned} \beta _{jk}(t) &= \Im \left [ p_{jk}(t) \left ( \frac {1}{\sqrt{a_{j}}} - i \frac{b_{j}}{\delta _{j}\sqrt{a_{j}}} \right ) + i q_{jk}(t) \frac{\sqrt{a_{j}}}{\delta _{j}} \right ] i \bar{\tau}', \end{aligned}$$

(20)

and based on (15), (18) and (16), we obtain

$$ T_{\mu }\equiv \alpha (t) \mu (t) - \beta (t) \int _{S} \frac{\mu (\tau )}{\tau -t}\,d\tau + \int _{S}K(t,\tau )\mu (\tau )\,d \tau = f(t), $$

(21)

where $\mu =(\mu _{1},\ldots ,\mu _{n})$ is search vector

$$ K(t,\tau ) = \frac{K^{1}(t,\tau )-K^{1}(t,t)}{\tau -t}, $$

(22)

and $n\times n$ matrix $K(t,\tau )$ is given by

$$\begin{array}{c}{K}_{jk}^1(t,\tau )={\overline{\tau }}^{\prime }(\tau -t)\mathrm{\Re }\left\{[p_{jk}(t)(\frac{1}{\sqrt{a_j}}-i\frac{b_j}{{\delta }_j\sqrt{a_j}})+i{q}_{jk}(t)][\frac{1}{\tau -t}-K_{0j}(t,\tau )]\right\},\hfill \\ \hfill K_{0j}(t,\tau )\equiv -\frac{{\overline{\tau }}_j^{\prime }}{t_j}-\frac{d{\overline{\tau }}_j}{d{s}_{{\tau }_j}}\frac{1}{{\tau }_j-t_j}-\frac{d\overline{\tau }}{d{s}_{\tau }}\frac{1}{\tau -t},j,k=1,\dots ,n.\end{array}$$

(23)

Due to (22) and (23) the last term in the left part of equality (21) represents a compact matrix integral operator and the second term a matrix singular integral operator, in which the integral is understood in the sense of Cauchy’s principal value. Likewise, due to (19) and (20) matrix coefficients $\alpha (t)$ and $\beta (t)$ satisfy Hölder’s condition.

Therefore, (21) is a system of singular integral equations equivalent to problem (17).

It is known that system (21) is normally solvable if the conditions

$$ \left \{ \begin{aligned} \det [\alpha (t) - \pi i \beta (t)] &\neq 0 \\ \det [\alpha (t) + \pi i \beta (t)] &\neq 0 \end{aligned} \right . $$

(24)

are fulfilled everywhere on $S$.

The index of this system is calculated by formula [14, 20]:

$$ \kappa = \frac {1}{2 \pi} \left [ \arg \frac {\det (\alpha + i \pi \beta )}{\det (\alpha - i \pi \beta )} \right ]_{S} $$

where $[]_{S}$ means the increment of the function in brackets at one circumvention of point $t$ around of curve $S$ from the fixed point $z_{0}=0$ on $S$ in the positive direction.

When fulfilling conditions (24) the well-known Noether theorems are used for the system of singular integral equations (21):

• 1. the corresponding (21) homogeneous system $T\mu _{0}=0$ and adjoin homogeneous system $T^{*}\mu _{0}=0$ have no more than finite number $l$ and $l'$ of linearly independent solutions,

  $$ l-l'=\kappa , $$

• 2. for the solvability of non-homogeneous system (21) it is necessary and sufficient that conditions

  $$ \int _{S}f(t)\mu _{*}^{(k)}(t)\, dt = 0,\qquad k=1,\ldots ,l', $$

  are fulfilled, where vectors $\{\mu _{*}^{(k)}\}$, $k=1,\ldots ,l'$, represent all linearly independent solutions of ad-joint homogeneous system $T^{*}\mu _{*}=0$.

 □

On the basis of equivalence between problem (17) and the system of singular integral equations (21), it is possible to conclude that for the solvability of this problem for arbitrary $f\in C^{0,h}(S)$ it is necessary and sufficient that $l'=0$, and the general solution to equation (21) has the following form

$$ \mu = \sum _{k=1}^{l} \beta _{k} \mu _{k} + \mu _{0} $$

where $\mu _{k}$ represents all linearly independent solutions of equation $T_{\mu _{0}}$. $\beta _{k}$ are real arbitrary constants and $\mu _{0}$ partial solution of the same equation.

If $\kappa =0$ and the homogeneous problem corresponding to (17) has only a trivial solution, then the non-homogeneous problem (17) has a solution which is the only one.

5 Conclusions

In this work, we have studied the exterior boundary value Poincaré problem for an elliptic system posed in equation (1) that satisfy the boundary conditions given by (2). We have offered some examples to show that the requirement of uniform ellipticity of system (1) does not guarantee the normal solvability (according to Noether). Nevertheless, we prove that, under certain conditions, it is possible to select from system (1) some decomposable types of partial differential equations (given by equation (14)) that are effectively solvable in the sense of Noether. For those systems, the Poincaré problem (equation (17)) is reduced by means of Theorem 1 to an equivalent system of singular integral equations.

The importance of integral equation methods in the solution of certain types of boundary value problems is universally accepted. From a practical point of view, the benefit may not always be very relevant when interior problems are concerned, but for exterior problems, where the region of interest is the infinite extent, an integral equation formulation may be virtually indispensable and our results are able to provide added value on this topic.

Appendix

The solution presented in equation (15) can be reformulated through a variable substitution:

$$\begin{gathered} x_{j}(x,y) = \frac {1}{\sqrt{a_{j}}} x, \qquad y_{j}(x,y) = \frac{\sqrt{a_{j}}}{\delta _{j}} y - \frac{b_{j}}{\delta _{j}\sqrt{a_{j}}} x, \qquad \delta _{j}^{2} = a_{j} c_{j} - b_{j}^{2}, \\ u_{j}(x,y) = \Re \varphi _{j}(z_{j}) = w_{j}(x_{j},y_{j}), \quad z_{j}=x_{j}+iy_{j} \end{gathered}$$

we canexpress it as:

$$ u_{j}(x,y) = w_{j}(x_{j}(x,y), y_{j}(x,y)) $$

where $w_{j}$ is a harmonic function corresponding to the real part of a analytical complex function $\varphi _{j}$.

By applying the chain rule, we can compute the following derivatives:

$$\begin{aligned} \frac{\partial ^{2} u_{j}}{\partial x^{2}} &= \frac{\partial ^{2} w_{j}}{\partial x_{j}^{2}}\frac{1}{a_{j}} - 2 \frac{\partial ^{2} w_{j}}{\partial x_{j}\partial y_{j}} \frac{b_{j}}{\delta _{j}a_{j}} + \frac{\partial ^{2} w_{j}}{\partial y_{j}^{2}} \frac{b_{j}^{2}}{\delta _{j}^{2}a_{j}}, \\ \frac{\partial ^{2} u_{j}}{\partial y^{2}} &= \frac{\partial ^{2} w_{j}}{\partial y_{j}^{2}} \frac{a_{j}}{\delta _{j}^{2}}, \\ \frac{\partial u_{j}}{\partial x\partial y} &= \frac{\partial ^{2} w_{j}}{\partial y_{j}^{2}} \frac{-b_{j}}{\delta _{j}^{2}} + \frac{\partial ^{2} w_{j}}{\partial x_{j}\partial y_{j}} \frac{1}{\delta _{j}}. \end{aligned}$$

By substituing the previous expressions into equation (14):

$$ a_{j} \frac{\partial ^{2} u_{j}}{\partial x^{2}} + 2 b_{j} \frac{\partial ^{2} u_{j}}{\partial x \partial y} + c_{j} \frac{\partial ^{2} u_{j}}{\partial y^{2}} = 0, $$

after some straightforward algebraic operations, it is equivalent to

$$ \frac{\partial ^{2} w_{j}}{\partial x_{j}^{2}} + \frac{\partial ^{2} w_{j}}{\partial y_{j}^{2}} = 0 $$

that holds because $w_{j}$ is a harmonic function.