Reports of the Academy of Sciences of the USSR

1. Volume 122, No. 4

MATHEMATICS

B. V. BOYARSKII

A GENERAL REPRESENTATION OF SOLUTIONS OF AN ELLIPTIC SYSTEM OF \(2n\) EQUATIONS IN THE PLANE

(Presented by Academician I. N. Vekua on 13 V 1958)

1. In the present work we consider systems of first-order equations of the form

\[ u_x=\widetilde A u_y+\widetilde B u+f, \tag{*} \]

where \(u\) is an unknown \(2n\)-component real vector; \(\widetilde A\) and \(\widetilde B\) are square matrices of order \(2n\), \(f\) is a real vector, given in some plane domain \(T\) of the variable \(z=x+iy\). The ellipticity of the system \((*)\) means that at every point of the domain \(T\) the equation
\[ \det(\widetilde A-\lambda E)=0 \]
has no real roots \((^1)\). We shall consider the system \((*)\) in the following complex canonical form:

\[ w_{\bar z}-Qw_z=Aw+B\bar w+F, \tag{1} \]

where \(w(z)\) is an \(n\)-component complex unknown vector; \(A\) and \(B\) are complex square matrices of order \(n\); \(F\) is a complex vector; \(Q\) is a quasidiagonal matrix of the form \(Q=\{Q_1,Q_2,\ldots,Q_p\}\), where \(Q_i=\{q_{lk,i}\}\) are square matrices, \(q_{lk,i}=0\) for \(k>l\), \(q_{ll,i}=q_i\), and \(q_{lk,i}=\beta_i^{\,l-k}\) for \(k<l\), \(q_i\) and \(\beta_i\) being complex functions defined in \(T\).

Equation (1) is the complex notation for the canonical form of the system \((*)\) in a real domain \((^{1,4,5})\). If \(T\) is simply connected, the elementary divisors of the matrix \(\widetilde A-\lambda E\) preserve their multiplicity in \(T\), the rank of each of the matrices \(\widetilde A-\lambda_iE\), where \(\det(\widetilde A-\lambda_iE)=0\), is constant in \(T\), and, in addition, the matrix \(\widetilde A\) is continuously differentiable in \(T\), then the system \((*)\) can be reduced to the form (1) in the entire domain \(T\) \((^{1,4})\).

By ellipticity, one has \(|q_i|<1\) in \(T\). If the system \((*)\) is assumed to be uniformly elliptic, i.e. \(|\operatorname{Im}\lambda_i|>\widetilde\lambda>0\) and \(|\lambda_i|\le M\) (\(\widetilde\lambda\) and \(M\) are positive constants), whenever \(\det(\widetilde A-\lambda_iE)=0\), then \(|q_i|\le q_0<1\), \(i=1,\ldots,p\), \(q_0=\mathrm{const}\); then it can be shown that in (1), without loss of generality, one may assume \(|\beta_i|<\beta_0\), \(i=1,\ldots,p\), where \(\beta_0\) is a prescribed arbitrarily small positive number; for sufficiently small \(\beta_0\) one has

\[ \max_i \sum_k |q_{ik}| \le q_0' <1,\qquad q_0'=\mathrm{const}. \tag{2} \]

In what follows we shall consider systems of the form (1) under the following assumptions: 1) the matrices \(A\), \(B\), and \(Q\) are given on the whole plane, with \(A=B=Q=0\) outside some sufficiently large circle \(K\); 2) \(Q\) possesses (generalized) derivatives \(Q_z\) and \(Q_z\in L_p(K)\), \(p>2\); \(A\) and \(B\) are measurable and bounded for \(z\in K\). We note that, using

inequality (2), a number of problems for the system (1) can be solved under \(A, B \subset L_p(K)\) and any measurable \(Q\) satisfying (2).

In the paper a theory of systems (1) is constructed, parallel to the theory of I. N. Vekua \((^2)\), who studied the case \(n=1\) and \(Q \equiv 0\). The case \(n>1\) differs from the case \(n=1\) in a number of respects; all of them are connected with the invalidity, in the general case, of Liouville’s theorem for systems (1) \((n>1)\).

By a regular solution of the system (1) we shall mean a vector \(w(z)\) possessing generalized derivatives \(w_z, w_{\bar z}\in L_p,\ p\geqslant 2,\) and satisfying equation (1) almost everywhere. An example is constructed of a system (1) which has a solution regular in the whole plane, vanishing at infinity and not identically equal to zero. However, in a number of cases Liouville’s theorem for systems (1) remains valid. This holds, for example, in the case when \(Q, A, B\) are constant inside \(K\) and are equal to zero outside \(K\).

2. Solutions of the system

\[ \Phi_{\bar z}-Q\Phi_z=0 \tag{3} \]

will be called \(Q\)-holomorphic vectors. The theory of a \(Q\)-holomorphic vector is completely analogous to the theory of a holomorphic vector. Under stronger assumptions than ours, a number of theorems of this theory are available in the work of Douglas \((^4)\). For a \(Q\)-holomorphic vector the generalized Liouville theorem is valid.

Theorem 1. There exists a matrix \(V(z,t)\), defined for all \(z\) and \(t\) and possessing the following properties:

1) \[ V'_{\bar z}-(Q'V')_z=0; \]

2) \[ V_{\bar t}-QV_t=0 \quad \text{for } z\ne t; \]

3) \[ \lim_{t\to\infty} tV(z,t)=\hat V(z);\quad \det \hat V(z)\ne0 \text{ for no } z,\text{ including } z=\infty; \]

4) \[ \lim_{z\to\infty} zV(z,t)=E \quad \text{for all } t; \]

5) \[ \lim_{\delta\to0}\int_{|z-t|=\delta} V(t,z)\,(dt+Q\overline{dt})\,w(t)=-2\pi i\,w(z) \]
for any finite \(z\) and for any continuous vector \(w(z)\).

Conditions 1), 2), 3), 4), 5) determine the matrix \(V(z,t)\) uniquely.

For the matrix \(V(z,t)\) the following expression can be indicated: denote by \(\zeta(z)\) the solution of equation (3) of the form
\[ \zeta(z)=zE-\frac{1}{\pi}\iint_K \frac{\omega(t)}{t-z}\,dK_t. \]
Using (2), it is not difficult to show that \(\zeta(z)\) exists and is unique. Further, for \(z\ne t\) there exists the matrix \((\zeta(t)-\zeta(z))^{-1}\). To determine \(V(z,t)\) we put
\[ V(z,t)=-(\zeta(t)-\zeta(z))^{-1}\varphi_z(z). \]
The matrix \(V(z,t)\) is naturally called the Cauchy kernel of equation (3). By \(\hat V(z,t)\) we shall denote the Cauchy kernel of the system
\[ -\varphi_{\bar z}+(Q'\varphi)_z=0, \]
conjugate to (3). The kernels \(V\) and \(\hat V\) are connected by the relation
\[ \hat V(z,t)=-V'(t,z). \]
The matrix \(V(z,t)\) plays the role of the matrix \((t-z)^{-1}E\) in the theory of a holomorphic vector.

For any vector \(w(z)\), continuous in a plane domain \(T\) bounded by a piecewise smooth contour \(L=L_0+L_1+\cdots+L_m\) and belonging to the Sobolev class \(W_p^1(T),\ p>1,\) the following generalization of the Pompeiu decomposition \((^{2,3})\), important in the theory of I. N. Vekua \((^2)\), holds:

\[ W(z)=\frac{1}{2\pi i}\int_L V(t,z)(dt+Q\overline{dt})\,w(t) -\frac{1}{\pi}\iint_T V(t,z)\,\omega(t)\,dT_t, \tag{4} \]

where \(\omega(z)=w_{\bar z}-Qw_z\).

By \(\overset{\circ}{V}(z)\) we shall denote the matrix solution of the equation

\[ \overset{\circ}{V}_{z}-(Q'\overset{\circ}{V}')_{z}=0 \quad\text{of the form}\quad \overset{\circ}{V}(z)=E-\frac{1}{\pi}\iint_{K}\frac{\omega(t)}{t-z}\,dK_t . \]

Such a solution always exists. With its aid one can define the \(Q\)-derivative \(\partial_z^{Q}w\) and the \(Q\)-integral \((Q)\int w\) of the vector \(w(z)\) by the formulas

\[ \partial_z^{Q}w=\overset{\circ}{V}^{-1}w_z,\qquad (Q)\int w=\int_{z_0}^{z}\overset{\circ}{V}(z)(dz+Q\,d\bar z)w(z). \]

It is not difficult to verify that the \(Q\)-derivative and the \(Q\)-integral of a \(Q\)-holomorphic vector are again a \(Q\)-holomorphic vector \((^4)\). The following lemma is important for applications to boundary-value problems:

Lemma 1. Every \(Q\)-holomorphic vector \(w(z)\), Hölder-continuous in the closed domain \(T+L\) \((L\) is a smooth contour\(),\) admits the representation

\[ w(z)=\frac{1}{\pi i}\int_L V(t,z)(\overline{dt}+Q\,\overline{dt})\mu(t)+iC, \]

where \(\mu(t)\) is a real vector, Hölder-continuous, and \(C\) is a real constant vector; the vector \(C\) is determined uniquely by the vector \(w(z)\); the vector \(\mu(t)\) is determined up to an additive constant on each of the contours \(L_j,\ j\geq 1\), bounding the domain \(T\) and lying inside the contour \(L_0\).

1. Formula (4) makes it possible to associate with system (1) the following system of Fredholm-type integral equations:

\[ w(z)+\frac{1}{\pi}\iint_K V(t,z)(Aw+B\bar w)\,dK_t=\Phi(z). \tag{5} \]

Equation (5) is equivalent to system (1), if \(\Phi\) is a \(Q\)-holomorphic vector. If \(w(z)\) is given, then the corresponding \(\Phi(z)\) is found by the formula

\[ \Phi(z)=\frac{1}{2\pi i}\int_L V(t,z)(dt+Q\,d\bar t)w(t). \]

For \(n=1\) and \(Q=0\) system (5) becomes the system of integral equations placed by I. N. Vekua at the foundation of his theory \((^2)\). As follows from the example in §1, the homogeneous equation (5) may, for \(n>1\), admit a finite number of nontrivial solutions \(w_k(z),\ k=1,2,\ldots,N\). Therefore the nonhomogeneous equation (5) is solvable only when a finite number of equalities of the form

\[ \operatorname{Re}\iint_K \Phi v_j\,dK=0,\quad j=1,2,\ldots,N, \]

are satisfied, where \(v_j\) are solutions of the homogeneous system of integral equations conjugate to (5) with respect to the scalar product

\[ [\Phi,v]=\operatorname{Re}\iint_K(\Phi,\bar V)\,dK,\qquad (\Phi,\psi)=\sum_{i=1}^{n}\Phi_i\psi_i . \]

By \(\psi_k(z),\ k=1,2,\ldots,N\), we shall everywhere denote regular nontrivial solutions vanishing at infinity of the system of differential equations conjugate to system (1):

\[ \psi_z-(Q'\psi)_z+A'\psi+\overline{B}'\bar\psi=0. \]

Theorem 2. Every solution of equation (1), bounded in the domain \(T\), is representable in the form

\[ w(z)=\Phi(z)+\iint_T \Gamma_1(z,t)\overline{\Phi(t)}\,dT +\iint_T \Gamma_2(z,t)\overline{\Phi(t)}\,dT +\sum_{k=1}^{N}c_k w_k(z), \tag{6} \]

where \(\Phi(z)\) is a \(Q\)-holomorphic vector; \(\Gamma_1(z,t)\), \(\Gamma_2(z,t)\) are matrices depending only on the domain \(T\) and on the coefficients of equation (1); \(c_k\) are real constants \((c_k=[w,\overline{w_k}])\). The vector \(\Phi(z)\) of representation (6) satisfies the conditions

\[ \operatorname{Im}\int_L \left(\Phi,\,(dt+Q'\overline{dt})\psi_k\right)=0, \qquad k=1,\ldots,N. \tag{7} \]

Conversely, if \(\Phi(z)\) is a bounded \(Q\)-holomorphic vector satisfying conditions (7), then formula (6) gives a solution of equation (1).

In addition to the kernels \(\Gamma_1(z,t)\) and \(\Gamma_2(z,t)\), we also consider the kernels

\[ \Omega_1(z,t)=V(t,z)+\iint_K \Gamma_1(z,\sigma)V(t,\sigma)\,dK_\sigma, \]

\[ \Omega_2(z,t)=\iint_K \Gamma_2(z,\sigma)\overline{V(t,\sigma)}\,dK_\sigma. \]

The kernels \(\Omega_1(z,t)\) and \(\Omega_2(z,t)\), as well as \(\Gamma_1(z,t)\) and \(\Gamma_2(z,t)\), are naturally continued to the whole plane in both variables \(z\) and \(t\). They vanish at infinity. For the kernels \(\Omega_1,\Omega_2,\Gamma_1\), and \(\Gamma_2\) one derives a number of integral and differential relations analogous to relations (5.17), (5.18), (5.19), (5.24), and (5.25) of the work \((^2)\). With the aid of the kernels \(\Omega_1\) and \(\Omega_2\), Cauchy’s formula for system (1) is written down.

Theorem 3. In order that the vector \(w(t)\), prescribed on \(L\) and Hölder-continuous on \(L\), be the boundary value of a regular solution of system (1) in the domain \(T\), Hölder-continuous in \(T+L\), it is necessary and sufficient that:

\[ 1)\quad \operatorname{Im}\int_L \left((dt+Q\overline{dt})w(t),\psi_k(t)\right)=0, \qquad k=1,2,\ldots,N. \]

\[ 2)\quad \frac{1}{2\pi i}\int_L \left[ \Omega_1(z,t)(dt+Q\overline{dt})w(t) -\Omega_2(z,t)\overline{(dt+Q\overline{dt})w(t)} \right] +\sum_{k=1}^{N} c_k w_k(z)=0 \]

for any point \(z\in \overline{T}+L\) and for some real constants \(c_1,c_2,\ldots,c_N\).

If the complement of the domain \(T=T^+\) is denoted by \(T^-\), then with the aid of the kernels \(\Omega_1(z,t)\) and \(\Omega_2(z,t)\) one can consider a Cauchy-type integral for system (1), write the usual Plemelj formulas and, in particular, give a solution of the following problem:

Find a piecewise-regular solution of system (1) satisfying the conditions

\[ w^+(t)-w^-(t)=\mu(t)\quad \text{on } L,\qquad w(\infty)=0. \tag{8} \]

Theorem 4. For the solvability of problem (8) it is necessary and sufficient that

\[ \operatorname{Im}\int_L \left(\psi_k,\,(dt+Q\overline{dt})\mu(t)\right)=0, \qquad k=1,2,\ldots,N. \tag{9} \]

Using the representations given above, one can transfer to solutions of system (1) a number of other propositions from the theory of I. N. Vekua. These results find application in the theory of boundary-value problems for system (1).

Received
11 V 1958

REFERENCES

\(^1\) I. G. Petrovsky, Lectures on Partial Differential Equations, Moscow—Leningrad, 1953.
\(^2\) I. N. Vekua, Matem. sborn., 31 (73), No. 2 (1952).
\(^3\) I. N. Vekua, DAN, 89, No. 5 (1953).
\(^4\) A. Douglis, Comm. Pure and Appl. Math., 6, 259 (1953).
\(^5\) V. V. Boyarskii, Proceedings of the Conference on the Theory of Functions of a Complex Variable, Moscow, 1958.