V. I. Shevchenko

On a Boundary-Value Problem for a Vector Holomorphic in a Half-Space

(Presented by Academician I. N. Vekua on 26 VII 1963)

In this note we show that the Riemann–Hilbert boundary-value problem for a vector holomorphic in a half-space is Fredholm, if the vector of the boundary condition does not lie in the tangent plane. The definition of a holomorphic vector is given in the works \((^1,{}^2)\).

Consider the problem \(\Gamma\):

Problem \(\Gamma\). Find a vector \(V\) with components \(p(x), u(x), v(x), w(x)\) (where \(x(x_1,x_2,x_3)\) is a point of the three-dimensional Euclidean space \(E_3\)), holomorphic in the half-space \(x_3>0\), vanishing at infinity, Hölder-continuous up to the boundary \(x_3=0\), and satisfying on the plane \(x_3=0\) the boundary conditions

\[ \lambda_{11}p+\lambda_{12}u+\lambda_{13}v+\lambda_{14}w=f_1, \]

\[ \lambda_{21}p+\lambda_{22}u+\lambda_{23}v+\lambda_{24}w=f_2, \tag{1} \]

where \(\lambda_{ij}(z)\) and \(f_i(z)\) \((i=1,2;\ j=1,2,3,4)\) are real functions of the point \(z(x_1,x_2)\) of the plane \(x_3=0\) of class \(C_\beta(E_2)\), \(0<\beta<1\), and the \(f_i(z)\) satisfy the condition \(f_i\in L_p^*\). The solution of the problem will naturally be sought in the class of functions whose boundary values belong to \(L_p(E_2)\), \(p>2/\beta\).

Denote by \(\Lambda_{ij}\) the determinant composed of the \(i\)-th and \(j\)-th columns of the matrix of the boundary condition, and let

\[ A(z)=\Lambda_{12}+\Lambda_{34};\qquad B(z)=\Lambda_{13}+\Lambda_{42};\qquad C(z)=\Lambda_{14}+\Lambda_{23}. \tag{2} \]

The vector with coordinates \((A,B,C)\) will be called the vector of the boundary condition.

In the particular case when the matrix of the boundary condition has the form

\[ \left\| \begin{array}{rrrr} g_1 & g_2 & g_3 & g_4\\ -g_4 & -g_3 & g_2 & g_1 \end{array} \right\| \]

and the coefficients \(g_i\) \((i=1,2,3,4)\) are constant, this problem was considered by A. V. Bitsadze \((^2)\). It was assumed there that \(C=g_1^2+g_2^2+g_3^2+g_4^2\ne0\).

We shall assume that the condition

\[ C(z)\ne0 \tag{3} \]

is satisfied everywhere, including the point at infinity.

Theorem. Under the assumptions made: 1) problem \(\Gamma\) is normally solvable; 2) the homogeneous problem and its adjoint have only a finite number of linearly independent solutions; 3) the index of problem \(\Gamma\) is equal to zero, i.e., the problem is Fredholm. (By the index we mean the index of the system of singular integral equations equivalent to problem \(\Gamma\).)

Condition (3) means that the vector of the boundary condition (2) does not lie in the tangent plane. For one second-order equation and a finite domain this result was obtained by Giraud. In one very special case

\[ \text{* The functions } \lambda_{ij}(z) \text{ have finite limits as } |z|\to\infty. \]

it to systems of equations of the 2nd order \((^3)\). The system that we consider can, generally speaking, be reduced to a system of equations of the 2nd order; however, in the inverse reduction one has to solve a problem equivalent to the original one. Moreover, in order to apply Giraud’s results the half-space must be reduced to a bounded domain. In doing so, singularities appear in the coefficients of the original system.

We shall rely on the known integral representation of a vector holomorphic in a half-space (see \((^2)\)):

\[ V(x)=\frac{1}{2\pi}\iint_{E_2}D\frac{1}{|x-\xi|}\,\gamma^3\mu(\xi)\,d\xi \tag{4} \]

(we use the notation from \((^8)\)), where \(\mu=(\mu_1,0,0,\mu_2)\) and \(\mu_1\) and \(\mu_2\) are the boundary values of the 1st and 4th components of \(V(x)\) on the plane \(x_3=0\).

Using the jump formulas for a two-dimensional integral of Cauchy type \((^2)\), we obtain a system of singular integral equations

\[ H\mu=f, \tag{5} \]

where \(\mu=(\mu_1,\mu_2)\), \(f=(f_1,f_2)\), and the operator \(H\) has the form

\[ H\mu \equiv L\mu+\frac{1}{2\pi}G\iint_{E_2}\hat D\frac{1}{|x-\xi|}\mu(\xi)\,d\xi . \]

The matrices \(L\), \(G\), and \(\hat D\) are given by the formulas

\[ L= \left\| \begin{array}{cc} \lambda_{11} & \lambda_{14}\\ \lambda_{21} & \lambda_{24} \end{array} \right\|, \qquad G= \left\| \begin{array}{cc} \lambda_{12} & \lambda_{13}\\ \lambda_{22} & \lambda_{23} \end{array} \right\|, \qquad \hat D= \left\| \begin{array}{cc} -\dfrac{\partial}{\partial x_2} & \dfrac{\partial}{\partial x_1}\\[4pt] \dfrac{\partial}{\partial x_1} & \dfrac{\partial}{\partial x_2} \end{array} \right\|. \]

By the uniqueness of representation (4), the original problem is completely equivalent to system (5).

We consider system (5) in the space \(L_p(E_2)\), \(p>2/\beta\). Let \(\Phi(z,y/|y|)\) be the symbol of the operator \(H\). Then \(\Phi=L+iGY\), where

\[ Y= \left\| \begin{array}{cc} -\dfrac{y_2}{|y|} & \dfrac{y_1}{|y|}\\[4pt] \dfrac{y_1}{|y|} & \dfrac{y_2}{|y|} \end{array} \right\|, \qquad y=(y_1,y_2)\in E_2 . \]

The symbol \(\Phi(z,y/|y|)\), obviously, belongs to the class \(C_\beta^\infty\) introduced in \((^4)\), and by definition \(H\) is a singular operator of class \(C_\beta^\infty\).

Since \(\det\Phi=C+iA\dfrac{y_1}{|y|}+iB\dfrac{y_2}{|y|}\) is nonzero by virtue of condition (3), there exists a matrix \(\Phi'\in C_\beta^\infty\) such that \(\Phi'\cdot\Phi=E\) (\(E\) is the identity matrix of the 2nd order).

Then, as is known, the operator \(H'\in C_\beta^\infty\) corresponding to the matrix \(\Phi'\) (see \((^4)\)) is a regularizer for the operator \(H\) (see \((^5)\)). Similarly it is shown that the adjoint operator \(H^*\) admits regularization. Assertions 1) and 2) of the theorem now follow from the theorems of § 2 of the book \((^5)\).

Let us show that the index \(\chi(H)\) of the operator \(H\) is equal to zero. Let, for definiteness, \(C(z)>0\). Consider the system of singular integral equations \(H_s\mu=f\), where

\[ L_s= \left\| \begin{array}{cc} g_1+s\lambda_1 & g_4+s\lambda_1\\ -g_4+s\lambda_4 & g_1-s\lambda_1 \end{array} \right\|; \qquad G_s= \left\| \begin{array}{cc} g_2+s\lambda_2 & g_3+s\lambda_3\\ -g_3+s\lambda_3 & g_2-s\lambda_2 \end{array} \right\|; \qquad 0\le s\le 1; \]

\[ g_1=\frac{\lambda_{11}+\lambda_{24}}{2}; \qquad g_2=\frac{\lambda_{12}+\lambda_{23}}{2}; \qquad g_3=\frac{\lambda_{13}-\lambda_{22}}{2}; \qquad g_4=\frac{\lambda_{14}-\lambda_{21}}{2}; \]

\[ \lambda_1=\frac{\lambda_{11}-\lambda_{24}}{2}; \qquad \lambda_2=\frac{\lambda_{12}-\lambda_{23}}{2}; \qquad \lambda_3=\frac{\lambda_{13}+\lambda_{22}}{2}; \qquad \lambda_4=\frac{\lambda_{14}+\lambda_{21}}{2}. \]

Let \(\Phi_s\) be the symbol of the operator \(H_s\). Then

\[ \det\Phi_s=C+(1-s^2)(\lambda_1^2+\lambda_2^2+\lambda_3^2+\lambda_4^2)>0 \]

and, obviously, \(H_s\in C_\beta^\infty\).

The elements of the symbolic matrix \(\Phi_{s+\Delta s}-\Phi_s\) and their derivatives up to the 4th order with respect to the coordinates of the point \(y\), \((y_1,y_2)\), as is easy to see, are small together with \(\Delta s\) for \(|y|\geqslant 1\), and, by Theorem 3 of paper \((^4)\), the norm \(\|H_{s+\Delta s}-H_s\|_p\) of the operator \(H_{s+\Delta s}-H_s\) in the space \(L_p,\ p>2/\beta\), is small together with \(\Delta s\).

By a known theorem (see \((^5)\), §2), \(\varkappa(H_{s+\Delta s})=\varkappa(H_s)\), i.e. the index of the operator \(H_s\) does not depend on \(s\). But for \(s=1\) we have the operator \(H\), and for \(s=0\) the operator \(H_0\). Consequently,

\[ \varkappa(H)=\varkappa(H_0). \]

This equality can also be obtained from a theorem of B. V. Boyarskii (see \((^6)\), appendix to Ch. II). The symbolic determinant of the operator \(H_0\) is
\[ \det\Phi_0=g_1^2+g_2^2+g_3^2+g_4^2=\rho^2>0. \]
Without loss of generality we may suppose that \(\rho(z)\equiv 1\). If this is not so, consider the operator \(P\mu=\dfrac{1}{\rho}\mu\). Let \(H_0'=PH_0\). Obviously, \(H_0'\in C_\beta^\infty\) and \(\varkappa(P)=0\). Passing to the symbolic matrices, we have
\[ \Phi_0'=\frac{1}{\rho}E\cdot\Phi_0 \]
and \(\det\Phi_0'=1\). In this case (see \((^5)\), §2)
\[ \varkappa(H_0')=\varkappa(P)+\varkappa(H_0)=\varkappa(H_0). \]

Consider now the system
\[ H_0\mu=f, \tag{6} \]
assuming that
\[ g_1^2+g_2^2+g_3^2+g_4^2=1. \tag{7} \]
By virtue of condition (7), there exist angles \(\theta(z),\varphi(z)\), and \(\psi(z)\), \(0\leqslant\theta(z)\leqslant\pi\); \(0\leqslant\varphi(z)\leqslant\pi\); \(0\leqslant\psi(z)<2\pi\), such that
\[ g_1=\cos\theta;\quad g_2=\sin\theta\cos\varphi;\quad g_3=\sin\theta\sin\varphi\cos\psi; \]
\[ g_4=\sin\theta\sin\varphi\sin\psi. \tag{8} \]

Let \(H_0^{(s)}\mu=f\) be a system of the form (6), but with coefficients \(g_i^{(s)}(z)\), \((i=1,2,3,4)\), defined by formulas (8) with the angles \(s\theta(z),\varphi(z),\psi(z)\), where \(0\leqslant s\leqslant 1\). The symbolic determinant of this system is identically equal to one. Similarly to what was done above, it is proved that \(\varkappa(H_0^{(s)})\) does not depend on \(s\). Consequently, the index of the system (6) coincides with the index of the system \(\mu=f\), i.e. is equal to zero. Therefore the index of the original system (5) is equal to zero in the space \(L_p(E_2),\ p>2/\beta\).

Let \(\mu_0\) be a solution of the homogeneous system (5), and let the operator \(H'\in C_\beta^\infty\) be the regularizer indicated above for the operator \(H\), i.e. \(H'H=I+T\), where \(I\) is the identity operator and \(T\) is a completely continuous operator in the space \(L_p,\ p>2/\beta\). Then \(\mu_0\) satisfies the equation \(\mu_0=-T\mu_0\).

It is easy to see that \(T\) is an integral operator with a weak singularity. From the properties of the operator \(T\) (see \((^7)\)) it follows that \(\mu_0\in C_\delta(E_2)\), \(\delta=\dfrac{p\beta-2}{p}\), and, consequently, \(\varkappa=0\) in \(C_\delta L_p(E_2)\), i.e. the problem \(\Gamma\) is Fredholm.

If the coefficients of the boundary condition (1), \(\lambda_{ij}\), are constant, then the problem \(\Gamma\) is always solvable, since, as the Fourier transform shows, the homogeneous problem \(\Gamma_0\) has only the trivial solution.

In conclusion I express my deep gratitude to Academician I. N. Vekua for valuable guidance in carrying out this work.

Novosibirsk State
University

Received
22 VII 1963

REFERENCES

1. Gr. C. Moisil, N. Theodoresco, Mathematica, 5, 141 (1931).
2. A. V. Bitsadze, Communications of the Academy of Sciences of the Georgian SSR, 16, No. 3 (1955).
3. G. Giraud, C. R., 192, 471 (1931).
4. A. P. Calderon, A. Zygmund, Am. J. Math., 79, 901 (1957).
5. S. G. Mikhlin, Multidimensional singular integrals and integral equations, Moscow, 1962.
6. B. V. Boyarskii, Doctoral dissertation, Moscow, 1961.
7. I. N. Vekua, Generalized analytic functions, Moscow, 1959.
8. V. I. Shevchenko, DAN, 146, No. 5 (1962).