MATHEMATICS

V. I. SHEVCHENKO

ON THE RIEMANN—HILBERT PROBLEM FOR A HOLOMORPHIC VECTOR

(Presented by Academician I. N. Vekua on February 2, 1965)

In this note we consider a boundary-value problem of Riemann—Hilbert type (problem A) for a holomorphic vector. A simple condition will be indicated that ensures the Noether property of problem A. Under this condition a formula will be obtained expressing the index of problem A. For this we use I. N. Vekua’s method of reducing a boundary-value problem to an equivalent system of singular integral equations. We note that the Riemann—Hilbert boundary-value problem for a holomorphic vector (in one special case) was first considered in the work of A. V. Bitsadze \((^1)\).

Let us first obtain an integral representation for a holomorphic vector. Let a closed Lyapunov surface \(S\) bound a domain \(G\) of three-dimensional space, homeomorphic to a ball, and let \(U(x)\) be the desired vector, holomorphic in the domain \(G\) and Hölder-continuous in \(G+S\). Denote by \(\nu\) the exterior normal to the surface \(S\) at the point \(\zeta\), and consider the Fredholm equation

\[ \mu_1(y)-\frac{1}{2\pi}\iint\limits_S \frac{\partial}{\partial \nu}\frac{1}{|y-\zeta|}\,\mu_1(\zeta)\,ds = 2f_1(y) \qquad (y\in S), \tag{1} \]

where \(f_1(y)\) are the boundary values of the first component of the vector \(U(x)\) on \(S\). It is well known (see \((^6)\)) that for a given function \(f_1(y)\) equation (1) always has, and moreover has uniquely, a solution \(\mu_1(y)\), Hölder-continuous on \(S\), if \(f_1(y)\) is Hölder-continuous.

Now consider the holomorphic vector defined by an integral of Cauchy type \((^{10})\)

\[ V(x)=\frac{1}{4\pi}\iint\limits_S D'\frac{1}{|x-\zeta|}\,DS_\zeta\,\mu(\zeta) \qquad (x\in G), \tag{2} \]

where the vector \(\mu(y)\) has components \([\mu_1(y),0,0,0]\).

Let \(W(x)=U(x)-V(x)\). Passing to the limit as \(x\to y\in S\), we obtain, by virtue of equation (1), that on the surface \(S\) the first component \(w_1\) of the vector \(W\) is equal to zero. But the components of a holomorphic vector are harmonic functions. Consequently, \(w_1(x)\equiv 0\) in the domain \(G\), so that \(W=[0,\widetilde w]\), where \(\widetilde w\) is a three-component vector satisfying in \(G\) the system \(\operatorname{div}\widetilde w=0\), \(\operatorname{rot}\widetilde w=0\). Therefore \(\widetilde w=\operatorname{grad}\varphi\), where \(\varphi(x)\) is a function harmonic in the domain \(G\), Hölder-continuous together with its first-order derivatives in \(G+S\).

Let \(g(x,\zeta)\) be the Green function of the domain \(G\) for the Neumann problem. From Green’s formula there follows the relation

\[ \varphi(x)=\iint\limits_S g(x,\zeta)\frac{\partial \varphi}{\partial \nu}\,ds_\zeta+\mathrm{const}, \]

so that \(\varphi(x)\) may be sought in the form

\[ \varphi(x)=\iint\limits_S g(x,\zeta) \left( \mu_2(\zeta)-\iint\limits_S \mu_2\,ds \right)ds+\mathrm{const}, \tag{3} \]

where \(\mu_2(y)\) is an unknown real function, Hölder-continuous on \(S\). Let \(\varphi(x)\equiv \mathrm{const}\) in \(G\). Differentiating formula (3) in the normal direction and passing to the limit as \(x\to y\in S\), we obtain
\[ \mu_2(y)=\iint_S \mu_2\,ds. \]
Consequently, the density \(\mu_2(y)\), up to an additive constant, is uniquely determined by the vector \(\operatorname{grad}\varphi\).

Thus every vector holomorphic in the domain \(G\) and Hölder-continuous in \(G+S\) can be represented in the form
\[ U(x)=\frac{1}{4\pi}\iint_S D'\frac{1}{|x-\xi|}DS_\xi[\mu_1(\xi),0,0,0]+ \]
\[ +\left[0,\operatorname{grad}\iint_S g(x,\xi)\left(\mu_2(\xi)-\iint_S\mu_2\,ds\right)ds\right], \tag{4} \]
where the first term is an integral of Cauchy type \((^{10})\), \(g(x,\xi)\) is the Green function of the domain \(G\) for the Neumann problem, and the function \(\mu_1(y)\) is uniquely determined by the vector \(U(x)\), while the function \(\mu_2(y)\) is determined up to a constant; hence the vector \(U(x)\) is trivial if and only if \(\mu_1\equiv 0\), \(\mu_2\equiv \mathrm{const}\).

Let us now consider the following boundary-value problem:

Problem A. Find a vector \(U(x)\) with components \([p,u,v,w]\), holomorphic in the domain \(G\), Hölder-continuous in \(G+S\), and satisfying on \(S\) the boundary condition
\[ \lambda_{k1}p+\lambda_{k2}u+\lambda_{k3}v+\lambda_{k4}w=f_k(y)\qquad (k=1,2), \tag{5} \]
where \(\lambda_{kj}(y)\) \((j=1,2,3,4)\) and \(f_k(y)\) are real functions of the point \(y\) of the surface \(S\), Hölder-continuous. Without loss of generality one may assume that
\[ (\lambda_{k1})^2+(\lambda_{k2})^2+(\lambda_{k3})^2+(\lambda_{k4})^2=1\qquad (k=1,2). \tag{6} \]

Denote by \(\Lambda_{kj}\) the determinant formed from the \(k\)-th and \(j\)-th columns of the matrix of the boundary condition (5), and let
\[ A(y)=\Lambda_{12}+\Lambda_{34};\qquad B(y)=\Lambda_{13}+\Lambda_{42};\qquad C(y)=\Lambda_{14}+\Lambda_{23}. \]
The vector \(l(y)\) with coordinates \((A(y),B(y),C(y))\) will be called the vector of the boundary condition.

Theorem. Let the domain \(G\) be the unit ball, and let the vector of the boundary condition not enter the tangent plane to the sphere \(S\). Then the boundary-value problem A is Noetherian; in particular, the homogeneous problem \(A^0\) has only a finite number \(k\) of linearly independent solutions, and for solvability of the nonhomogeneous problem A with right-hand side (5) it is necessary to impose a finite number \(k'\) of orthogonality conditions.

Suppose, in addition, that the coefficients of the boundary condition (5) are twice continuously differentiable* with respect to the (Cartesian) coordinates of the point \(y\) and that condition (6) holds. Then the integer \(k-k'=\varkappa\) (the index of problem A)
\[ \varkappa=-1. \tag{7} \]

Proof. First of all, note that for the ball the function \(g(x,\xi)\) has the form (see \((^8)\))
\[ g(x,\xi)=\frac{1}{4\pi}\left\{\frac{1}{|x-\xi|}+\frac{1}{|\xi|\cdot|x-\xi^*|}-\frac12\left(|\xi|^2+|x|^2\right)-\right. \]
\[ \left.-\ln\left(1-x\cdot\xi+|\xi|\cdot|x-\xi^*|\right)\right\}, \]
where \(\xi^*=\xi/|\xi|^2\) and \(x\cdot\xi=x_1\xi_1+x_2\xi_2+x_3\xi_3\).

Representing now the sought vector \(U(x)\) through the unknown densities \(\mu_1(y)\) and \(\mu_2(y)\) by formula (4) and passing to the boundary, we obtain, by virtue of

* Note added in proof. From the stability of the index under small changes of the coefficients it follows that it suffices to impose the requirement that the coefficients of the boundary condition be Hölder-continuous.

conditions (5), for determining the vector \(\mu(\mu_1,\mu_2)\), a system of two singular integral equations

\[ H\mu=f \quad \{f=(f_1,f_2)\}, \tag{8} \]

where the operator \(H\mu\) is easily written out explicitly.

Let us write the symbolic matrix \(\Phi(r)\) of the system (8):

\[ \Phi(r)= \left\| \begin{array}{cc} \lambda_{11}(y)+i\lambda_1\cdot\rho & \lambda_1\cdot n+i\lambda_1\cdot r\\ \lambda_{21}(y)+i\lambda_2\cdot\rho & \lambda_2\cdot n+i\lambda_2\cdot r \end{array} \right\|, \tag{9} \]

where \(r\) and \(n\) are, respectively, the unit tangent vector and the outer normal to the sphere \(S\) at the point \(y\), \(\rho=n\times r\). The determinant of the matrix (9), \(l(y)\cdot(n+ir)\), is nonzero by the hypothesis of the theorem, so that the resulting system of singular integral equations is Noetherian (see (7)). Let us note that a necessary and sufficient condition for solvability of problem A can also be formulated in terms of the adjoint problem \(A'\) \((^9)\).

Let us now consider the vector \(\lambda\) with components \([\lambda_{11},\lambda_1\cdot\rho,\lambda_1\cdot n,\lambda_1\cdot r]\), determined by the first row of the matrix (9). Choosing \(\rho,n,r\) as basis vectors of a new coordinate system with center at the point \(y\), we obtain
\((\lambda_1\cdot\rho)^2+(\lambda_1\cdot n)^2+(\lambda_1\cdot r)^2=|\lambda_1|^2=\lambda_{12}^2+\lambda_{13}^2+\lambda_{14}^2\), whence, in view of condition (6), it follows that \(|\lambda|=1\).

Let us compute the index of the system (8).

It is known \((^5,^4,^2)\) that the index of the system (8) differs only by a constant factor from the rotation \(\sigma(\lambda)\) of the vector \(\lambda\), and for the sphere this factor is equal to one. A formula for computing \(\sigma(\lambda)\) is given in \((^2)\). By means of elementary transformations of the determinant standing under the integral sign, we obtain

\[ \sigma(\lambda)= -\frac{1}{\pi} \iint\limits_S \left| \begin{array}{cc} \dfrac{\partial}{\partial\theta}\lambda_{11} & \dfrac{\partial}{\partial\theta}(\lambda_1\cdot\nu)\\[6pt] \dfrac{\partial}{\partial\varphi}\lambda_{11} & \dfrac{\partial}{\partial\varphi}(\lambda_1\cdot\nu) \end{array} \right| \,d\theta\,d\varphi . \]

By Green’s formula, \(\sigma(\lambda)=0\).

Among the solutions of the homogeneous system (8) there certainly occurs the vector \((0,a)\), \(a=\mathrm{const}\ne0\). In view of the correspondence between \(U\) and \(\mu\), determined by formula (4), \(\varkappa=-1+\sigma(\lambda)\), whence formula (7) follows.

Although formula (7) has been proved for a ball, using the arguments given above one can show that it is valid for an arbitrary domain \(G\), homeomorphic to a ball, bounded by a three-times continuously differentiable surface \(S\) (\(\theta\) and \(\varphi\) in this case denote intrinsic coordinates on the surface). Here, of course, it is assumed that the vector of the boundary condition \(l(y)\) does not enter the tangent plane to the surface \(S\). The Noetherian property of problem A under this condition was proved in \((^9)\).

Let us further note that a formula for the index of the problem with boundary condition (5) (even for more general systems of the form \(DU=\varepsilon BU\) \((^9)\), where the values of the parameter \(\varepsilon\) do not belong to a certain discrete set) can be obtained starting from the accompanying boundary problems \(A_*\) and \(A^*\) \((^3)\).

In conclusion I express my deep gratitude to Acad. I. N. Vekua for his constant attention to the work.

Novosibirsk State University

Received
10 XII 1964

CITED LITERATURE

\(^1\) A. V. Bitsadze, Reports of the Academy of Sciences of the Georgian SSR, 16, No. 3 (1955).
\(^2\) B. V. Boyarskii, Bull. de l’Acad. polon. sci. S. math., 9, No. 10 (1963).
\(^3\) I. N. Vekua, Generalized Analytic Functions, Moscow, 1959.
\(^4\) A. I. Volpert, DAN, 142, No. 4 (1962).
\(^5\) S. G. Mikhlin, Materials for the Joint Soviet-American Symposium on Partial Differential Equations, Novosibirsk, 1963.
\(^6\) S. G. Mikhlin, Lectures on Linear Integral Equations, Moscow, 1959.
\(^7\) S. G. Mikhlin, Multidimensional Singular Integrals and Integral Equations, Moscow, 1962.
\(^8\) S. L. Sobolev, Equations of Mathematical Physics, Moscow, 1954.
\(^9\) V. I. Shevchenko, Candidate’s dissertation, Novosibirsk, 1964.
\(^10\) V. I. Shevchenko, DAN, 153, No. 6 (1963).