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Reports of the Academy of Sciences of the USSR
- Vol. 146, No. 5
MATHEMATICS
V. I. SHEVCHENKO
ON A LOCAL HOMEOMORPHISM OF THREE-DIMENSIONAL SPACE REALIZED BY A SOLUTION OF A CERTAIN ELLIPTIC SYSTEM
(Presented by Academician I. N. Vekua on 10 VII 1962)
In the present paper we consider a certain analogue of the complex Beltrami equation in three-dimensional space
\[ DU-Q\bar D U=0, \tag{1} \]
where \(U(x)\) is an unknown four-component real vector, and the operators \(D\) and \(\bar D\) are formed with the aid of the matrices
\[ \gamma_1= \begin{Vmatrix} 0&1&0&0\\ 1&0&0&0\\ 0&0&0&-1\\ 0&0&1&0 \end{Vmatrix},\qquad \gamma_2= \begin{Vmatrix} 0&0&1&0\\ 0&0&0&1\\ 1&0&0&0\\ 0&-1&0&0 \end{Vmatrix},\qquad \gamma_3= \begin{Vmatrix} 0&0&0&1\\ 0&0&-1&0\\ 0&1&0&0\\ 1&0&0&0 \end{Vmatrix} \]
and have the form
\[ D=\sum_{i=1}^{3}\gamma_i\frac{\partial}{\partial x_i},\qquad \bar D=\sum_{i=1}^{3}\bar\gamma_i\frac{\partial}{\partial x_i}, \]
(the bar above denotes transposition).
\(Q(x)\) is a real matrix of order four,
\[ Q(x)=\sum_{i=1}^{3}\bar\gamma_i q_i(x), \]
defined at each point \(x=(x_1,x_2,x_3)\) of a finite domain \(G\) of three-dimensional Euclidean space \(E_3\). The equation \(DU=0\) was considered by A. V. Bitsadze, who was interested mainly in boundary-value problems (see \((^2)\)).
Suppose that the system (1) is elliptic in the sense of Petrovsky in the closed domain \(G\). Then
\[ \det\left(\sum_{i=1}^{3}\Gamma_i\xi_i\right) =-(1-r^4)(\xi_1^2+\xi_2^2+\xi_3^2)^2, \]
where \(\Gamma_i=\gamma_i-Q\bar\gamma_i\), and from ellipticity there follows the requirement
\[ r(x)=\sqrt{q_1^2(x)+q_2^2(x)+q_3^2(x)}\le q_0<1,\qquad q_0=\mathrm{const}. \tag{2} \]
Regarding \(Q(x)\) we shall assume that the \(q_i(x)\) possess generalized derivatives \(\partial q_i/\partial x_k\) \((i,k=1,2,3)\) in the sense of S. L. Sobolev, with \(\partial q_i/\partial x_k\in L_p(\bar G)\), \(p>3\), and that the inequality (2) is fulfilled. Hence, in particular, it follows that \(Q\in C_\alpha(\bar G)\). In what follows, by \(\partial/\partial x_k\) we shall always mean a generalized derivative and shall consider, generally speaking, generalized solutions of equation (1).
Using the ideas of the book \((^1)\), we shall show that equation (1) always admits a solution, any three components of which realize a local homeomorphism of the space \(E_3\) onto the space defined by these components.
Consider the operator
\[ T\omega=-\frac{1}{4\pi}\iint_G \bar D\,\frac{1}{|x-\xi|}\,\omega(\xi)\,d\xi . \]
Here \(\omega\) and \(T\omega\) are four-component vectors.
If \(\omega\in L_p(\bar G)\), \(p>1\), then, as B. V. Boyarskii showed, the function \(\psi(x)\equiv T\omega\) has (generalized) derivatives \(\partial\psi(x)/\partial x_s\in L_p(\bar G)\), and
\[ \frac{\partial\psi(x)}{\partial x_s} = -\frac{1}{4\pi}\,{}^{*}\!\iint_G \frac{\partial}{\partial x_s} \left(\bar D\,\frac{1}{|x-\xi|}\right)\omega(\xi)\,d\xi +\frac{1}{3}\bar\gamma_s\omega(x),\qquad x\in G, \tag{3} \]
where the integral is understood in the sense of the principal value, and the estimate holds \(({}^3,{}^4)\)
\[ L_p\left(\frac{\partial\psi}{\partial x_s}\right)\leq B_p L_p(\omega), \tag{4} \]
where \(B_p\) does not depend on \(\omega\) (we use the notation of \(({}^1)\)).
Introduce the operators
\[ \Pi_s\omega\equiv \frac{\partial\psi}{\partial x_s},\qquad s=1,2,3; \]
\[ \Pi\omega\equiv g(x)=\sum_{s=1}^{3}\bar\gamma_s\Pi_s\omega . \]
From (3), for \(\omega\in L_p(\bar G)\), there follows the representation \((x\in G)\)
\[ \Pi\omega = -\frac{1}{4\pi}\,{}^{*}\!\iint_G \bar D^{\,2}\,\frac{1}{|x-\xi|}\,\omega(\xi)\,d\xi +\frac{1}{3}\sum_{s=1}^{3}\bar\gamma_s^{\,2}\omega, \]
and if \(\omega\in C_\alpha(\bar G)\), then
\[ \Pi\omega = -\frac{1}{4\pi}\iint_G \bar D^{\,2}\,\frac{1}{|x-\xi|} [\omega(\xi)-\omega(x)]\,d\xi +\Phi(x)\omega(x), \tag{5} \]
where
\[ \Phi(x)=\frac{1}{4\pi}\int_S \bar D S_\xi\cdot \bar D\,\frac{1}{|x-\xi|}. \]
Here \(S\) is a Lyapunov surface bounding the domain \(G\), and
\[ \bar D S_\xi=\sum_{i=1}^{3}\bar\gamma_i\alpha_i\,dS_\xi; \]
\(\alpha_i\) are the direction cosines of the exterior normal to \(S\).
From inequality (4) it follows that
\[ L_p(\Pi\omega)\leq C_p L_p(\omega). \tag{6} \]
We shall need the following properties of the operators \(T\omega\) and \(\Pi\omega\). Let \(\omega\in L_p(\bar G)\), \(p>3\). Then \((x,y\in E_3)\)
\[ |\psi(x)-\psi(y)|\leq M L_p(\omega)|x-y|^\alpha,\qquad \alpha=\frac{p-3}{p}. \tag{7} \]
If \(\omega\in C_\alpha(\bar G)\) and the boundary of the domain \(G\) is sufficiently smooth, then
\[ T\omega\in C_\alpha^1(\bar G); \tag{8} \]
\[ |g(x)-g(y)|\leq M_\alpha' H(\omega,\alpha)|x-y|^\alpha \quad (x,y\in G), \tag{9} \]
i.e. \(\Pi\omega\in C_\alpha(\bar G)\) and
\[ C_\alpha(\Pi\omega)\leq M_\alpha C_\alpha(\omega). \tag{10} \]
These relations are proved analogously to the plane case (see \(({}^1)\)). For \(\Pi_s\omega\) the same estimates hold as for \(\Pi\omega\).
We shall seek a solution of equation (1) in the form
\[ U=Z+T\omega,\qquad \omega\in L_2(\overline G), \tag{11} \]
where
\[ Z= \begin{Vmatrix} x_1+x_2+x_3\\ x_2\\ x_3\\ x_1 \end{Vmatrix}. \]
It is easy to compute that \(DZ=0\), \(\overline DZ=2\begin{Vmatrix}0\\1\\1\\1\end{Vmatrix}=\mu\).
From (3) it follows that \(DT\omega=\omega\). Substituting (11) into (1), we obtain for \(\omega\) the singular integral equation
\[ \omega-Q\Pi\omega=Q\mu. \tag{12} \]
We shall show that equation (12), as a linear equation in \(L_2(\overline G)\), is always solvable.
It is proved that \(C_2=1\), i.e. the norm of the operator \(\Pi\omega\) in the space \(L_2(\overline G)\) is equal to one. Then
\(L_2(Q\Pi\omega)\le q_0 L_2(\Pi\omega)\le q_0L_2(\omega)\le L_2(\omega)\), and, by the contraction mapping principle, equation (1) has a unique solution \(U(x)\) of the form (11) in the space \(L_2(\overline G)\).
Reducing equation (1) to an integral equation with the operator \(T\omega\) alone, it is not difficult to show, using (7) and \(\partial q_i/\partial x_k\in L_p(\overline G)\), \(p>3\), that the constructed solution \(U(x)\in C_\beta(\overline G)\) for some \(\beta\), \(0<\beta<1\).
Let us formulate the main theorem:
Theorem 1. Let \(G_0\) be a neighborhood of some fixed point \(x_0\) and let \(Q\in C_\alpha(\overline G_0)\). Suppose, moreover, that
\[ |q_1(x_0)|+|q_2(x_0)|+|q_3(x_0)|<1. \tag{13} \]
Then in some sufficiently small neighborhood \(G'_0\) \((G'_0\subset G_0)\) of the point \(x_0\) there exists a solution of equation (1), any three components of which realize a local homeomorphism of the space \(E_3\) into the space determined by these components. This solution belongs to the class \(G_\alpha^1(\overline G'_0)\), \(0<\alpha<1\).
Let us outline the proof of the theorem. Without loss of generality, one may take the point \(x_0\) to be the origin of coordinates. Denote \(q_1(0)=a\), \(q_2(0)=b\), \(q_3(0)=c\), and in equation (1) perform a change of the unknown functions by setting \(AU=V\), where \(A=e-a\gamma_1-b\gamma_2-c\gamma_3\). Since \(\det A=1-r_0^4\), \(r_0^2=a^2+b^2+c^2\), it follows from (2) that the matrix \(A^{-1}\) exists. Equation (1) then takes the form
\[ DV=Q_1\widetilde DV=0, \tag{14} \]
where \(\widetilde DV=\overline D A^{-1}V\) and \(Q_1(0)=0\).
Let \(G_\delta\) be the closed ball with center at the origin. Denote by \(C_\alpha^0(G_\delta)\) the set of vectors \(\omega\in C_\alpha(E_3)\) that vanish outside \(G_\delta\) and satisfy the additional condition \(\omega(0)=0\).
A solution of equation (14) in the ball \(G_\delta\) is sought in the form (11), where \(\omega\in C_\alpha^0(G_\delta)\), which leads to the singular integral equation for \(\omega\)
\[ \omega-Q_1\widetilde\Pi\omega=Q_1\nu, \tag{15} \]
where \(\widetilde\Pi\omega=\overline D A^{-1}T\omega\) and \(\nu=\overline D A^{-1}Z\).
Using (9) and (10), as well as the estimate for \(Q_1(x)\) in the ball \(G_\delta\), we prove that for sufficiently small fixed \(\delta\) equation (15) has, moreover, a unique solution in the space \(C_\alpha(G_\delta)\).
By virtue of (8), the constructed solution \(V \in C_\alpha^1(G_\delta)\). Therefore equation (1) has in the ball \(G_\delta\) the solution
\[ U=A^{-1}(Z+T\omega), \tag{16} \]
belonging to the same class \(C_\alpha^1(G_\delta)\).
Denote by \(A_j^{-1}\) the matrix of size \(3\times 4\) which is obtained from the matrix \(A^{-1}\) by deleting the \(j\)-th row. Then the vector
\[ U_j=A_j^{-1}(Z+T\omega)\qquad (j=1,2,3,4) \]
will have three components, among which the \(j\)-th component of the vector \(U\) is not included. The Jacobian \(\Delta_j\) of each transformation \(U_j(x)\) of the space \(E_3\) is computed directly and estimated with the aid of inequalities (2) and (13). For sufficiently small fixed \(\delta\), all four Jacobians \(\Delta_j\) \((j=1,2,3,4)\) in the ball \(G_\delta\) are simultaneously different from zero.
Theorem 2. Let \(U(x)=(p(x),u(x),v(x),w(x))\) be a holomorphic vector in the domain \(G\) (see (2)) and let \(\Phi(U)\equiv\Phi(p,u,v,w)\) be a smooth vector whose components \(\Phi^1,\Phi^2,\Phi^3,\Phi^4\) depend on \(p,u,v,w\). In order that the vector \(\Phi(U(x))\) be holomorphic for every \(U(x)\), it is necessary and sufficient that \(\Phi\) satisfy the system of equations:
\[ \begin{aligned} \Phi_p^1&=\Phi_u^2=\Phi_v^3=\Phi_w^4,\\ \Phi_p^2&=-\Phi_u^1=-\Phi_v^4=\Phi_w^3,\\ \Phi_p^3&=\Phi_u^4=-\Phi_v^1=-\Phi_w^2,\\ \Phi_p^4&=-\Phi_u^3=\Phi_v^2=-\Phi_w^1. \end{aligned} \tag{17} \]
Conditions (17) are the monogeneity conditions for the quaternionic function \(\hat{\Phi}(\hat{U})=\Phi^1+i\Phi^2+j\Phi^3+k\Phi^4\) of the quaternionic argument \(\hat{U}=p+iu+jv+kw\) (see (5)). But, as shown in (6), every such function has the form \(\hat{\Phi}(\hat{U})=\hat{U}M+N\), where \(M\) and \(N\) are constant quaternions. Therefore, for any holomorphic \(U\), only the vector
\[ \Phi(U)=BU+\Phi_0, \tag{18} \]
where \(\Phi_0\) is a constant vector and the matrix \(B\) has the form
\[ B= \begin{Vmatrix} m_1 & -m_2 & -m_3 & -m_4\\ m_2 & m_1 & m_4 & -m_3\\ m_3 & -m_4 & m_1 & m_2\\ m_4 & m_3 & -m_2 & m_1 \end{Vmatrix}. \tag{19} \]
Let now \(Q\) be a constant matrix in the domain \(G\). Then, together with the smooth solution (11), equation (1) will have the solution
\[ W(x)=BA^{-1}(Z+T\omega)+W_0, \]
where \(B\) is an arbitrary matrix of the form (19) and \(W_0\) is a constant vector.
In conclusion, I express my deep gratitude to Academician I. N. Vekua for his constant attention to this work.
Novosibirsk State University
Received
6 VII 1962
CITED LITERATURE
- I. N. Vekua, Generalized Analytic Functions, 1959.
- A. V. Bitsadze, Reports of the Academy of Sciences of the Georgian SSR, 16, No. 3 (1955).
- B. V. Boyarskii, Dissertation, Moscow State University, 1955.
- A. Calderon, A. Zygmund, Acta Math., 88, 85 (1952).
- N. M. Krylov, DAN, 55, No. 9 (1947).
- A. S. Meilikhzon, DAN, 59, No. 3 (1948).