Abstract Generated abstract
The paper studies a three-dimensional analogue of the Beltrami equation, formulated as an elliptic first-order system for a four-component real vector with matrix coefficients satisfying a uniform bound. Using singular integral operators and estimates in Lebesgue and Hölder spaces, the equation is reduced to an integral equation whose solvability follows from contraction arguments under appropriate regularity assumptions on the coefficients. The main result proves that near a point where the coefficient norm is sufficiently small, there exists a \(C^1_\alpha\) solution such that any three of its four components define a local homeomorphism of three-dimensional space. The paper also characterizes transformations preserving holomorphic vectors in this setting, relating them to affine quaternionic-linear maps.
Full Text
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).