MATHEMATICS

V. I. SHEVCHENKO

CONSTRUCTION OF LOCAL AND GLOBAL HOMEOMORPHISMS FOR ONE CLASS OF EQUATIONS IN QUATERNIONS

(Presented by Academician I. N. Vekua, June 13, 1963)

It is known that in the plane case \((n = 2)\) the Beltrami equation
\(W_{\bar z} - q(z)W_z = 0\) always has homeomorphic solutions (for more detail see \(\left({}^{1}\right)\)). Of interest is the question of the existence of homeomorphic solutions (homeomorphisms) for the spatial Cauchy—Riemann system \((n \geqslant 3)\) and for elliptic systems related to it.

It is known that the number of components of a holomorphic vector (i.e., a solution of the Cauchy—Riemann system) coincides with the dimension of the space (and with the number of equations) only in the cases \(n = 2, 4, 8\). Since under homeomorphisms the dimension of the space remains invariant, the Cauchy—Riemann system admits solutions carrying out topological mappings only in two-, four-, and eight-dimensional spaces. In particular, in the three-dimensional case the Cauchy—Riemann system, generally speaking, has no homeomorphic solutions (the holomorphic vector has 4 components). However, as shown in \(\left({}^{4}\right)\), for an elliptic system related to the Cauchy—Riemann conditions there always exists a solution, any three components of which carry out a local homeomorphism of the space \(E_3\) onto the space determined by these components.

In the present note we show the existence of a global homeomorphism for an elliptic system of special form in four-dimensional space. For the proof we make extensive use of the ideas of the book \(\left({}^{1}\right)\). The eight-dimensional case can be considered analogously.

Consider the system

\[ \sum_{k=1}^{4} A_k(x)\frac{\partial}{\partial x_k}U = 0, \tag{1} \]

where \(U(x)\) is an unknown four-component real vector, \(A_k(x)\), \(k = 1,2,3,4\), are matrices of quaternion type, i.e.,

\[ A_k(x) = a_{k1}(x)e + a_{k2}(x)i + a_{k3}(x)j + a_{k4}(x)k, \]

where the matrices \(i, j, k\) have the form

\[ i = \begin{Vmatrix} 0 & 1 & 0 & 0\\ -1 & 0 & 0 & 0\\ 0 & 0 & 0 & -1\\ 0 & 0 & 1 & 0 \end{Vmatrix}, \qquad j = \begin{Vmatrix} 0 & 0 & -1 & 0\\ 0 & 0 & 0 & -1\\ 1 & 0 & 0 & 0\\ 0 & 1 & 0 & 0 \end{Vmatrix}, \qquad k = \begin{Vmatrix} 0 & 0 & 0 & -1\\ 0 & 0 & 1 & 0\\ 0 & -1 & 0 & 0\\ 1 & 0 & 0 & 0 \end{Vmatrix}, \]

\(e\) is the identity matrix of order four. The \(a_{kl}(x)\) are real functions of the point \(x(x_1, x_2, x_3, x_4)\) of a finite domain \(G\) of four-dimensional Euclidean space \(E_4\), bounded by a twice continuously differentiable surface.

If system (1) is elliptic, then it can always be written in the form (by changing, perhaps, the independent variables)

\[ D_1U - Q_2D_2U - Q_3D_3U - Q_4D_4U = 0, \tag{2} \]

where the matrices \(Q_l(x)\), \(l=1,2,3,4\), are easily expressed in terms of \(A_k(x)\), \(k=1,2,3,4\), and the operators \(D_l\) have the form:

\[ D_1=e\frac{\partial}{\partial x_1}+i\frac{\partial}{\partial x_2}+j\frac{\partial}{\partial x_3}+k\frac{\partial}{\partial x_4}, \]

\[ D_2=e\frac{\partial}{\partial x_1}+i\frac{\partial}{\partial x_2}-j\frac{\partial}{\partial x_3}-k\frac{\partial}{\partial x_4}, \]

\[ D_3=e\frac{\partial}{\partial x_1}-i\frac{\partial}{\partial x_2}+j\frac{\partial}{\partial x_3}-k\frac{\partial}{\partial x_4}, \]

\[ D_4=e\frac{\partial}{\partial x_1}-i\frac{\partial}{\partial x_2}-j\frac{\partial}{\partial x_3}+k\frac{\partial}{\partial x_4}. \]

For simplicity we shall restrict ourselves to considering the system

\[ D_1U-QD_2U=0, \tag{3} \]

where

\[ Q(x)=eq_1(x)+iq_2(x)+jq_3(x)+kq_4(x), \]

although all the results are valid for the more general system (2).

Suppose that system (3) is elliptic in the sense of Petrovsky. This means (see \((^4)\)) that everywhere in the domain \(G\)

\[ (1-r^2)^2+4q_2^2\ne 0,\qquad r^2(x)=\sum_{i=1}^{4}q_i^2(x). \]

It is natural to assume that the stronger inequality is satisfied

\[ r(x)\le q_0<1,\qquad q_0=\text{const}. \tag{4} \]

Inequality (4) makes it possible to construct, for system (3), generalized solutions of the class \(W_2^{(1)}\), assuming only the measurability of the functions \(q_i(x)\). The general scheme for constructing homeomorphic solutions is as follows.

Using the idea of I. N. Vekua (see \((^1)\)), we seek a solution of system (3) in the form (cf. \((^4)\))

\[ \mathcal W(x)=Z+T\omega, \tag{5} \]

where

\[ Z= \begin{Vmatrix} x_1\\ x_2\\ x_3\\ x_4 \end{Vmatrix}, \qquad T\omega=-\frac{1}{4\pi^2}\iint_G \overline{D_1}\frac{1}{|x-\xi|^2}\,\omega(\xi)\,d\xi, \]

which gives, for the four-component vector \(\omega\), the singular integral equation

\[ \omega-Q\Pi_2\omega=Q_1, \tag{6} \]

where \(\Pi_2\omega\equiv D_2T\omega\) and \(Q_1=QD_2Z\).

To prove the solvability of equation (6), the contraction mapping principle is used in one of the functional spaces \(L_2\), \(L_p\), \(W_p^{(1)}\), \(p>4\), or \(C_\alpha\).

Theorem 1. Let \(Q(x)\in C_\alpha(\overline G)\). If system (3) is elliptic in the sense of Petrovsky, then in some sufficiently small neighborhood \(G_0\) of any point \(x_0\) of the domain \(G\) there exists a solution \(\mathcal W_0(x)\) of system (3), which realizes a local homeomorphism of the space \(E_4\) onto the space \(W_4\) defined by the components of the solution. Moreover, \(\mathcal W_0(x)\in C_\alpha^1(\overline G_0)\).

Analogously \((^1)\), theorems can be proved concerning the smoothness of the constructed solution \(\mathcal W_0(x)\) depending on the smoothness of the matrix \(Q(x)\).

Theorem 2. If \(Q(x) \in W_p^{(1)}(\overline{G})\), \(p>4\), and the norm of the matrix \(Q(x)\) in this space is sufficiently small, i.e. the quantities \(\|q_i(x)\|_{W_p^{(1)}}\), \(i=1,2,3,4\), are small, then there always exists a global homeomorphism \(\mathcal W(x)\) of system (3), i.e. there exists a four-component vector function \(\mathcal W(x)\), satisfying system (3) everywhere in \(\overline{G}\) and mapping the domain \(G\) onto some domain \(G_W\) of the four-dimensional space \(W_4\). This mapping is \(\varepsilon\)-quasiconformal and belongs to \(C_\alpha^1(\overline{G})\), where \(\alpha=\dfrac{p-4}{p}\). If, moreover, \(Q(x) \in W_p^{(1)}(E_4)\), \(p>4\), with \(Q(x) \equiv 0\) outside some sufficiently large ball, and the norm of the matrix \(Q(x)\) in this space is still sufficiently small, then there exists a complete homeomorphism \(\mathcal W_*(x)\) of system (3) from the space \(E_4\) onto the space \(W_4\). In this case
\[ \mathcal W_*(x)-Z \in C_\alpha^1(E_4), \qquad \alpha=\frac{p-4}{p}, \]
and \(\mathcal W_*(x)\) realizes an \(\varepsilon\)-quasiconformal mapping of the space \(E_4\) onto the space \(W_4\).

For the proof, the matrix \(Q(x)\) is extended, with preservation of its class, to the whole space, in such a way that \(Q(x) \equiv 0\) in a neighborhood of infinity, and a solution \(\mathcal W(x)\) of the form (5) is constructed whose Jacobian is everywhere different from zero. By virtue of the conditions on \(Q(x)\), it possesses sufficient smoothness. To show the one-to-one character of the constructed solution, we use the Lefschetz–Hopf theorem on the algebraic number of fixed points of a vector field (see \((^3)\)); hence the mapping \(\mathcal W(x)\) is topological and therefore, by Brouwer’s theorem (see \((^3)\)), maps the domain \(G\) onto some domain \(G_W\) of the space \(W_4\).

I consider it my pleasant duty to express deep gratitude to Acad. I. N. Vekua for posing the problem and for guidance of the work.

Novosibirsk State
University

Received
8 VI 1963

References

\({}^1\) I. N. Vekua, Generalized Analytic Functions, Moscow, 1959.
\({}^2\) S. L. Sobolev, Some Applications of Functional Analysis in Mathematical Physics, Publishing House of the Siberian Branch of the USSR Academy of Sciences, 1962.
\({}^3\) P. S. Aleksandrov, Combinatorial Topology, Moscow–Leningrad, 1947.
\({}^4\) V. I. Shevchenko, DAN, 146, No. 5 (1962).