A. I. VOLPERT

ON THE INDEX OF SYSTEMS OF TWO-DIMENSIONAL SINGULAR INTEGRAL EQUATIONS

(Presented by Academician I. G. Petrovskii, 20 IX 1961)

In the present paper a formula is derived for the index of systems of two-dimensional singular integral equations, as well as a formula for the index of boundary-value problems for a system of harmonic functions in a three-dimensional domain. The latter is a generalization of the formula obtained in (¹).

1. We consider a system of singular integral equations

\[ a(x)u(x)+\int_S b(x,y-x)u(y)\,d_y S+Tu=f(x), \tag{1} \]

where \(S\) is a triply smooth surface, homeomorphic to a sphere, bounding a certain three-dimensional domain \(D\); \(x,y\) are points on \(S\); \(a(x)\) is a complex square matrix of order \(p\), defined and continuous on \(S\) and having continuous derivatives along \(S\) up to the second order; \(b(x,\alpha)\) is a complex square matrix of order \(p\), defined and continuous for \(x\in S\) and arbitrary nonzero vectors \(\alpha=(\alpha_1,\alpha_2,\alpha_3)\); \(T\) is a regular integral operator. The following conditions are assumed to be fulfilled: 1) \(b(x,\alpha)\) has continuous derivatives of arbitrary order with respect to the coordinates of the point \(\alpha\), and is twice continuously differentiable with respect to \(x\) along \(S\); 2) \(b(x,\rho\alpha)=\rho^{-2}b(x,\alpha)\) \((\rho>0,\alpha\ne0)\), as well as the condition for the existence of the singular integral in (1) (see (²)). The solution \(u\) of system (1) is sought in the space \(L^2(H)\) of functional columns of height \(p\), whose elements are square-summable along \(S\) (satisfy a Hölder condition on \(S\)), and the right-hand side \(f\) is assumed to belong to the same spaces.

With system (1), in the well-known way (see (²)), there is associated a symbol \(\Phi(\tau)\)—a square matrix of order \(p\), defined and continuous on the set \(P\) of all unit tangent vectors \(\tau\) to \(S\). As is known (see (²)), if the condition

\[ \det \Phi(\tau)\ne 0 \quad (\tau\in P) \tag{2} \]

is satisfied, system (1) is normally solvable; the subspaces of solutions of the homogeneous system (1) \((f=0)\) and of the homogeneous adjoint system are finite-dimensional, and the difference \(\varkappa=k-k^*\) between the dimensions \(k\) and \(k^*\) of these subspaces is called the index of system (1). S. G. Mikhlin proved that for \(p=1\) the index is equal to zero. In (¹) it was established that for \(p>1\) the index is, generally speaking, different from zero. In Theorem 1 a formula will be obtained for the index of system (1) for \(p>1\).*

To each matrix \(\Phi(\tau)\), defined and continuous on \(P\) and satisfying condition (2), there is associated an integer \(l\), defined as follows. Let first \(p=2\), and let \((\Phi_1(\tau)+i\Phi_2(\tau),\Phi_3(\tau)+i\Phi_4(\tau))\) be one of the rows of the matrix \(\Phi(\tau)\); \(\varphi(\tau)=(\Phi_1(\tau),\Phi_2(\tau),\Phi_3(\tau),\Phi_4(\tau))\). \(\psi(\tau)=\varphi(\tau)/|\varphi(\tau)|\), where \(|\varphi|\) is the length of the vector \(\varphi\). \(\psi\) maps \(P\) into the unit—

* The results presented below were reported by the author at the Fourth All-Union Mathematical Congress in July 1961.

three-dimensional sphere. By definition, \(l(\Phi)\) is the degree of this mapping. Obviously, \(l(\Phi)\) does not depend on the arbitrary choice of the row of the matrix \(\Phi\). For \(p>2\), as is known, the matrix \(\Phi(\tau)\) can be continuously deformed, preserving condition (2), into the matrix
\[ \begin{pmatrix} E & 0\\ 0 & \Phi_0(\tau) \end{pmatrix}, \]
where \(E\) is the identity matrix of order \(p-2\), and \(\Phi_0(\tau)\) is a matrix of second order. In this case \(l(\Phi_0)\) is uniquely determined by the matrix \(\Phi\). By definition \(l(\Phi)=l(\Phi_0)\).

When condition (2) is satisfied, the following holds.

Theorem 2. The index \((\varkappa)\) of system (1) is computed by the formula
\[ \varkappa = l(\Phi), \tag{3} \]
where \(\Phi\) is the symbol of this system.

1. Consider the following boundary-value problem: find in the domain \(D\) a solution \(v\) of the system \(\Delta v=0\), satisfying the boundary condition
   \[ \lim_{x\to y} B\left(y,\frac{\partial}{\partial x}\right)v(x)=f(y) \qquad (x\in D,\ y\in S), \tag{4} \]
   where \(\Delta\) is the Laplace operator;
   \[ B\left(y,\frac{\partial}{\partial x}\right)=\sum_{j=1}^{3} B_j(y)\frac{\partial}{\partial x_j}; \]
   \(B_j(y)\) \((j=1,2,3)\) are complex square matrices of order \(p\), defined on \(S\) and having second continuous derivatives along \(S\); \(f(y)\in H\). The solution \(v\) is sought in the class of functional columns of \(p\) elements, having second continuous derivatives in \(D\) and first continuous derivatives in \(D+S\). It is assumed that the condition of Ya. B. Lopatinskii \((^3)\) is fulfilled:
   \[ \det B(x,\nu(x)+i\tau)\ne 0 \tag{5} \]
   for all \(x\in S\) and unit vectors \(\tau\) tangent to \(S\) at the point \(x\). Here \(\nu(x)\) is the unit normal vector to \(S\) at the point \(x\). It is known that for \(p=1\) the index of problem (4) is equal to zero. For \(p>1\), the following holds.

Theorem 2. The index \((\varkappa)\) of problem (4) is computed by the formula
\[ \varkappa = l(B), \tag{6} \]
where \(B\) is the matrix occurring in (5).

1. Proof of the theorems. Consider the group \(G\) of all invertible square matrices of order \(p\), defined and continuous on \(P\), which are symbols of systems of the form (1). To each \(\Phi\in G\) associate an integer \(\varkappa(\Phi)\), the index of the corresponding system (1). If \(\Phi\in G\) and \(\Phi_1\in G\) can be continuously deformed into one another while preserving condition (2), then \(\varkappa(\Phi)=\varkappa(\Phi_1)\). Therefore from \(l(\Phi)=0\) it follows that \(\varkappa(\Phi)=0\), and since \(l\) and \(\varkappa\) homomorphically map \(G\) into the additive group of integers, for any \(\Phi\in G\)
   \[ \varkappa(\Phi)=\gamma l(\Phi), \tag{7} \]
   where \(\gamma\) is a certain constant.

We reduce problem (4) to a system of singular integral equations by means of the simple-layer potential. The symbol of this system is equal to \(B(x,\nu(x)+i\tau)\), and therefore, on the basis of (7), the index \((\varkappa)\) of problem (4) is \(\varkappa=\gamma l(B)\). For a complete proof of the theorems it remains only to show that \(\gamma=1\), and this, evidently, is sufficient to do for \(p=2\). As the boundary operator in (4), take an operator defined by an elliptic first-order system on \(S\) with characteristic \(\varkappa=0\) (see \((^1)\)). Then, as shown in \((^1)\), \(\varkappa=2\). It is verified directly that in this case \(l(B)=2\), whence it follows that \(\gamma=1\).

Institute of Chemical Physics
Academy of Sciences of the USSR

Received
15 IX 1961

REFERENCES CITED

\(^1\) A. I. Volpert, DAN, 133, No. 1, 13 (1960).
\(^2\) S. G. Mikhlin, UMN, 3, no. 3, 29 (1948).
\(^3\) Ya. B. Lopatinskii, Ukr. Math. Zhurn., 5, No. 2, 123 (1953).