UDC 517.946.9

MATHEMATICS

V. I. SHEVCHENKO

ON THE HILBERT PROBLEM FOR A HOLOMORPHIC VECTOR

(Presented by Academician I. N. Vekua, December 3, 1965)

In this note we consider the problem of linear conjugation (the Hilbert problem) for a holomorphic vector (briefly, problem H). We shall show that, under certain conditions on the conjugation matrix (G), problem H is Fredholm. In addition, by introducing the adjoint problem H′ into consideration, we obtain a necessary and sufficient condition for the solvability of problem H. We note that the Hilbert problem for a holomorphic vector was first considered by A. V. Bitsadze ((^1)) in the case when the constant matrix (G) (see formula (2) below) has the form

[
G_0=
\begin{Vmatrix}
g_1 & g_2 & g_3 & g_4\
-g_2 & g_1 & -g_4 & g_3\
-g_3 & g_4 & g_1 & -g_2\
-g_4 & -g_3 & g_2 & g_1
\end{Vmatrix}.
\tag{1}
]

We investigate problem H by means of singular integral equations, following the ideas of Chapter IV of the book by I. N. Vekua ((^4)). The adjoint problem H′ is introduced analogously to the way this was done by B. V. Boyarskii ((^3)) in the planar case.

A vector (U(x)) satisfying a certain elliptic system (DU=0) (see ((^2))) is called holomorphic. A vector (V(x)) satisfying the system (D'V=0) (the prime denotes transposition) will be called co-holomorphic.

Let (S) be a Lyapunov surface, homeomorphic to a sphere, lying in Euclidean space (E_3). Consider problem H (see ((^1))): to find a piecewise-holomorphic vector (U(x)) satisfying the boundary condition on (S):

[
U^+(y)=G(y)U^-(y)+f(y),
\tag{2}
]

where (G(y), f(y)\in C_\alpha(S)), (0<\alpha<1).

Denote by (M_{ij}^{kl}) the minor composed of the elements of the (k)-th and (l)-th rows and the (i)-th and (j)-th columns of the matrix (G(y)), and let

[
\begin{aligned}
\Gamma_{11}={}&M_{13}^{13}+M_{13}^{42}+M_{42}^{42}+M_{42}^{13}
+M_{14}^{23}+M_{14}^{14}+M_{23}^{23}+M_{23}^{14},\
\Gamma_{22}={}&M_{12}^{12}+M_{12}^{34}+M_{34}^{12}+M_{34}^{34}
+M_{14}^{23}+M_{14}^{14}+M_{23}^{23}+M_{23}^{14},\
\Gamma_{33}={}&M_{12}^{12}+M_{12}^{34}+M_{34}^{12}+M_{34}^{34}
+M_{13}^{42}+M_{13}^{13}+M_{42}^{42}+M_{42}^{13},\
2\Gamma_{12}=2\Gamma_{21}={}&-\bigl(M_{12}^{13}+M_{12}^{42}+M_{34}^{13}+M_{34}^{42}
+M_{13}^{12}+M_{13}^{34}+M_{42}^{12}+M_{42}^{34}\bigr),\
2\Gamma_{13}=2\Gamma_{31}={}&-\bigl(M_{12}^{14}+M_{12}^{23}+M_{34}^{14}+M_{34}^{23}
+M_{14}^{12}+M_{14}^{34}+M_{23}^{12}+M_{23}^{34}\bigr),\
2\Gamma_{23}=2\Gamma_{32}={}&-\bigl(M_{13}^{14}+M_{13}^{23}+M_{42}^{14}+M_{42}^{23}
+M_{14}^{13}+M_{14}^{42}+M_{23}^{42}+M_{23}^{13}\bigr).
\end{aligned}
]

Everywhere in this note we shall assume that the following condition (\Gamma) is fulfilled: all principal minors (including the determinant) of the matrix with elements (\Gamma_{ij}) ((i,j=1,2,3)) are positive on (S). We note that for matrices of the form (1) condition (\Gamma) coincides with the condition (\det G(y)\ne 0).

With the help of an integral of Cauchy type (see (2)), problem (H) is reduced to an equivalent system of singular integral equations for the unknown vector (\mu(y)):

[
L\mu \equiv (G+E)\mu+(E-G)P\mu=2f,
\tag{3}
]

where

[
P\mu=\frac{1}{2\pi}\iint\limits_S D'\frac{1}{|y-\xi|}\,DS_\xi\mu(\xi),\qquad
y\in S,\qquad
DS_y=\left(\sum_{i=1}^{3}\alpha_i\gamma_i\right)ds_y^{*},
]

(\alpha_i) are the direction cosines of the exterior normal to (S) at the point (y), and (E) is the identity matrix of order four.

We shall consider equation (3) in the space (L_p(S)) for some (p>1). Let (\sigma(L)) be the symbolic matrix of the operator (L). A direct calculation shows that

[
\operatorname{Re}\det\sigma(L)=4\sum_{i,j=1}^{3}\Gamma_{ij}\tau_i\tau_j,
\tag{4}
]

where ((\tau_1,\tau_2,\tau_3)) is a unit tangent vector to the surface (S) at the point (y). From condition (\Gamma) it follows that the quadratic form (4) is positive definite, so that

[
\det\sigma(L)\ne 0.
\tag{5}
]

From the results of S. G. Mikhlin (see ((^5)), § 40) it follows that the operator (L) is Noetherian in the space (L_p(S)), i.e., first, the homogeneous equation (3) and the adjoint homogeneous equation

[
L^{}\chi\equiv (G'+E)\chi+P^{}(E-G')\chi=0
\tag{6}
]

have only finite numbers (k) and (k') of linearly independent solutions and, second, for the solvability of equation (3) it is necessary and sufficient that

[
\iint\limits_S \chi_j'(y)f(y)\,ds_y=0
\qquad (j=1,2,\ldots,k'),
\tag{7}
]

where (\chi_1,\ldots,\chi_{k'}) is a complete system of linearly independent solutions of equation (6). By virtue of condition (5), any solution from the space (L_q(S)), (q>1), of equations (3) and (6) belongs to (C_\alpha(S)) (see ((^7))); thus the operator (L) is Noetherian also in the space (C_\alpha(S)).

Let us introduce the adjoint homogeneous problem (H_0'): to find a piecewise-cogolomorphic vector (V(x)) satisfying the boundary condition on (S):

[
G^{*}(y)V^{+}(y)=V^{-}(y),
\tag{8}
]

where

[
G^{*}(y)=\left(\sum_{i=1}^{s}\alpha_i\gamma_i\right)G'(y)\left(\sum_{i=1}^{3}\alpha_i\gamma_i\right).
]

With the help of an integral of Cauchy type for a cogolomorphic vector, problem (H_0') is reduced to an equivalent system of singular integral equations:

[
M\nu\equiv (G^{}+E)\nu+(G^{}-E)Q\nu=0,
\tag{9}
]

where

[
Q\nu=\frac{1}{2\pi}\iint\limits_S D\frac{1}{|y-\xi|}\,D'S_\xi\nu(\xi).
]

Solutions of equations (6) and (9) can be connected by the relation

[
\nu=\left(\sum_{i=1}^{3}\alpha_i\gamma_i\right)(G'-E)\chi.
\tag{10}
]

[
\text{* }\gamma_i\ (i=1,2,3)\text{ are constant matrices of order four, with the help of which the operator }
D=\sum_{i=1}^{3}\gamma_i\frac{\partial}{\partial x_i}\text{ is formed (see (2)).}
]

From formulas (10) and (6) we obtain

[
2\chi=-\left(\sum_{i=1}^{3} a_i'\gamma_i\right)(v+Qv).
\tag{11}
]

Therefore the problem (\mathrm{H}0') has exactly (k') linearly independent solutions (V_1(x),\ldots,V(x)). From formulas (7) and (11) it follows:

Theorem 1. For the solvability of the nonhomogeneous boundary-value problem (\mathrm{H}), it is necessary and sufficient that, for every solution (V_j(x)) ((j=1,\ldots,k')) of the adjoint homogeneous problem (\mathrm{H}_0'), the condition

[
\iint_S {v_j^+(y)}'\,DS_y f(y)=0
\tag{12}
]

be satisfied.

The integer (\varkappa(G)=k-k') will be called the index of the boundary-value problem (\mathrm{H}) generated by the matrix (G).

Consider the problem (\mathrm{H}) generated by the matrix (G_t(y)), depending on the parameter (t), (0\le t\le 1),

[
G_t(y)=tG(y)+(1-t)G_0(y),
]

where (G_0(y)) is a matrix of the form (1) with elements

[
\begin{aligned}
g_1&=\tfrac14(g_{11}+g_{22}+g_{33}+g_{44}),\
g_2&=\tfrac14(g_{12}-g_{21}+g_{43}-g_{34}),\
g_3&=\tfrac14(g_{13}-g_{31}+g_{24}-g_{42}),\
g_4&=\tfrac14(g_{14}-g_{41}+g_{32}-g_{23}).
\end{aligned}
\tag{13}
]

Let the operator (L_t) correspond to the matrix (G_t(y)) by formula (3). Then, by virtue of equality (4),

[
\operatorname{Re}\det\sigma(L_t)=\operatorname{Re}\det\sigma(L)+4(1-t^2)\sum_{i=1}^{4}\left(\sum_{j=1}^{3}\lambda_{ij}\tau_j\right)^2>0,
\tag{14}
]

where by (\lambda_{ij}(y)) are denoted certain linear combinations of the elements of the matrix (G(y)) of the type of formulas (13). In particular, for (t=0) formula (14) gives (g_1^2(y)+g_2^2(y)+g_3^2(y)+g_4^2(y)>0). Since (\varkappa(G)) coincides with the index of the Noether operator (L), (\varkappa(G)) does not change under continuous (in Hölder’s sense) deformations of the matrix (G(y)) preserving condition (5) (see (5), § 37, § 2 and (6)). Therefore (\varkappa(G)=\varkappa(G_0)), and, without loss of generality, one may assume (see (6)) that

[
g_1^2(y)+g_2^2(y)+g_3^2(y)+g_4^2(y)=1.
\tag{15}
]

The boundary condition of the problem (\mathrm{H}_0') for a matrix of the form (1) satisfying condition (15) has the form

[
V^+(y)=\widetilde{G}(y)V^-(y),
\tag{16}
]

where

[
\widetilde{G}(y)=
\begin{pmatrix}
g_1 & -g_2 & -g_3 & -g_4\
g_2 & g_1 & -g_4 & g_3\
g_3 & g_4 & g_1 & -g_2\
g_4 & -g_3 & g_2 & g_1
\end{pmatrix}.
]

Let (V(v_1,v_2,v_3,v_4)) be a cogolomorphic vector satisfying condition (16). Then (U(-v_1,v_2,v_3,v_4)) will be a solution of the problem (\mathrm{H}_0), and conversely. Consequently, (k=k') and (\varkappa(G_0)=0). Thus, we have proved

Theorem 2. If condition (\Gamma) is fulfilled, the boundary-value problem (\mathrm{H}) is Fredholm.

Formulating the problem (\mathrm{H}) for the upper and lower half-spaces (x_3>0) and (x_3<0) ((S=E_2)), one must additionally require that (U^-(y))

and (f(y)\in L_p(E_2)), (p>1). Theorems 1 and 2 remain valid* in this case as well.

Let us connect the Hilbert problem with the Riemann–Hilbert problem for the half-plane (problem (\Gamma)) (see ((^6))). In ((^6)) problem (\Gamma) is reduced to an equivalent system of singular integral equations

[
A\hat\mu+B\hat P\hat\mu=\hat f,\qquad
\hat\mu=(\mu_1,\mu_2),\quad \hat f=(f_1,f_2),
\tag{17}
]

a condition for the Noetherian property of this problem is indicated, and it is shown that, in computing the index of system (17), without loss of generality one may assume that

[
A(y)=
\left|
\begin{matrix}
g_1 & g_4\
-g_4 & g_1
\end{matrix}
\right|,
\qquad
B(y)=
\left|
\begin{matrix}
g_2 & g_3\
-g_3 & g_2
\end{matrix}
\right|,
]

where condition (15) is satisfied. The author takes this opportunity to correct an inaccuracy he made in ((^6)) in proving the fact that (\chi(H_0)=0). The complete space (E_2) is homeomorphic to the two-dimensional sphere (S_2), so that one may assume that the functions (g_1,g_2,g_3,g_4) realize a (continuous) mapping of the sphere (S_2) into the sphere (15) (S_3). However, every such mapping is unstable (see ((^8))). Therefore, by an arbitrarily small change of the functions (g_1,g_2,g_3,g_4) (the index of the corresponding Noether operator does not change), one can ensure that this mapping omits any point of (S_3), for example the point ((-1,0,0,0)). Then the homotopy indicated in ((^6)), which it is convenient to rewrite in the equivalent form
(g_1^{(s)}(z)=\cos s\theta,\quad g_k^{(s)}(z)=g_k(z)\sin s\theta/\sin\theta\ (k=2,3,4),\quad 0\le s\le1,)
shows that system (17) is Fredholm.

The same assertion is not difficult to obtain from Theorem 2. Indeed, add to system (17) the equations (\mu_3=f_3) and (\mu_4=f_4), and denote by (\hat H\mu) the operator generated by the left-hand side of the resulting system. Let (\hat L) be the operator corresponding, by formula (3), to the matrix (\hat G_0) of the form (1), where (g_2) and (g_3) are taken with the opposite sign. Then (R\hat L R^*\mu=\hat H\mu+K\mu), where (K\mu) is some completely continuous operator in the space (L_p(E_2)), (p>1), while the simplest operator (R) is uniquely recovered from its symbolic matrix

[
\frac12
\left|
\begin{matrix}
1 & -i\tau_2 & i\tau_1 & 0\
0 & i\tau_1 & i\tau_2 & 1\
-1 & -i\tau_2 & i\tau_1 & 0\
0 & -i\tau_1 & -i\tau_2 & 1
\end{matrix}
\right|
\qquad
(\tau_1^2+\tau_2^2=1,\quad i^2=-1).
]

Since the operator (R) is invertible, (\chi(\hat H)=\chi(\hat L)) in the space (C_\alpha L_p(E_2)), (p>1).

In conclusion, I express my deep gratitude to Academician I. N. Vekua for his constant attention to this work.

Donetsk Computing Center
of the Academy of Sciences of the Ukrainian SSR

Received
23 XI 1965

CITED LITERATURE

(^1) A. V. Bitsadze, DAN, 93, No. 4 (1953).
(^2) A. V. Bitsadze, Reports of the Academy of Sciences of the Georgian SSR, 16, No. 3 (1955).
(^3) B. V. Boyarskii, Doctoral dissertation, Moscow, 1961.
(^4) I. N. Vekua, Generalized Analytic Functions, Moscow, 1959.
(^5) S. G. Mikhlin, Multidimensional Singular Integrals and Integral Equations, Moscow, 1962.
(^6) V. I. Shevchenko, DAN, 154, No. 2 (1964).
(^7) V. I. Shevchenko, DAN, 163, No. 2 (1965).
(^8) V. Gurevich, G. Volman, Dimension Theory, Moscow, 1948.

* With the obvious modification of condition (\Gamma).