V. S. Vinogradov

ON AN ANALOGUE OF THE CAUCHY–RIEMANN SYSTEM IN FOUR-DIMENSIONAL SPACE

(Presented by Academician N. N. Bogolyubov, 18 VII 1963)

Let us consider the system of differential equations

\[ \begin{aligned} &\frac{\partial u_1}{\partial x_1} -\frac{\partial u_2}{\partial x_2} -\frac{\partial u_3}{\partial x_3} -\frac{\partial u_4}{\partial x_4}=0,\\ &\frac{\partial u_1}{\partial x_2} +\frac{\partial u_2}{\partial x_1} -\frac{\partial u_3}{\partial x_4} +\frac{\partial u_4}{\partial x_3}=0,\\ &\frac{\partial u_1}{\partial x_3} +\frac{\partial u_2}{\partial x_4} +\frac{\partial u_3}{\partial x_1} -\frac{\partial u_4}{\partial x_2}=0,\\ &\frac{\partial u_1}{\partial x_4} -\frac{\partial u_2}{\partial x_3} +\frac{\partial u_3}{\partial x_2} +\frac{\partial u_4}{\partial x_1}=0, \end{aligned} \tag{1} \]

where \(u_1, u_2, u_3, u_4\) are real functions of the four independent variables \(x_1, x_2, x_3, x_4\). The system (1) is elliptic, since the corresponding form

\[ D(\xi_1,\xi_2,\xi_3,\xi_4)= \left| \begin{array}{rrrr} \xi_1 & -\xi_2 & -\xi_3 & -\xi_4\\ \xi_2 & \xi_1 & -\xi_2 & \xi_3\\ \xi_3 & \xi_4 & \xi_1 & -\xi_2\\ \xi_4 & -\xi_3 & \xi_2 & \xi_1 \end{array} \right| = (\xi_1^2+\xi_2^2+\xi_3^2+\xi_4^2)^2 \]

is strictly positive.

In what follows we shall use quaternion notation for the system (1). Let

\[ U=u_1+i u_2+j u_3+k u_4 \]

be a quaternion with multiplication laws for the units
\(i^2=j^2=k^2=-1,\ ij=-ji=k\). Then the system (1) may be written in the form

\[ \partial_x U= \left( \frac{\partial}{\partial x_1} +i\frac{\partial}{\partial x_2} +j\frac{\partial}{\partial x_3} +k\frac{\partial}{\partial x_4} \right) (u_1+i u_2+j u_3+k u_4)=0. \tag{2} \]

We shall study solutions of the system (2) in a certain bounded domain \(D\) with smooth boundary \(S\), which at each point has a continuously varying normal \(n\). Denote by
\(\alpha=\alpha_1+i\alpha_2+j\alpha_3+k\alpha_4\) the quaternion whose components are the direction cosines of the normal, \(\alpha_i=\cos n\cdot x_i\); then the Gauss–Ostrogradsky formula is written in our notation as follows:

\[ \iiint\!\!\int_D \partial_x U\, dv = \iiint_S \alpha U\, ds, \tag{3} \]

where \(dv\) and \(ds\) are, respectively, the elements of volume and surface.

Since the operation of multiplication for quaternions is noncommutative, we obtain two different differential operators (systems) when multiplying \(U\) by \(\partial_x\) on the right and on the left; multiplication on the left by \(\partial_x\) is understood according to formula (2), while for multiplication of \(U\) by \(\partial_x\) on the right one must formally multiply them as quaternions and then set \(u_i\partial/\partial x_j=\partial u_i/\partial x_j\).

In what follows, in various combinations we shall denote by \(|\partial_x U|\) the action of \(\partial_x\) only on \(U\) from the left, and by \([U\partial_x]\) the action of \(\partial_x\) only on \(U\) from the right.

Take two quaternion-valued functions \(U(x)\) and \(V(x)\) (\(x\) denotes \((x_1,x_2,x_3,x_4)\)), continuously differentiable in the domain \(D\), and consider

operator

\[ [V\,\partial_x]\,U+V[\partial_x U]. \tag{4} \]

It is easily verified that

\[ \int_D \{[V\,\partial_x]\,U+V[\partial_x U]\}\,dv= \]

\[ =\int_D \left\{ \frac{\partial}{\partial x_1}(VU)+ \frac{\partial}{\partial x_2}(ViU)+ \frac{\partial}{\partial x_3}(VjU)+ \frac{\partial}{\partial x_4}(VkU) \right\}\,dv =\int_S V\alpha U\,ds. \]

Thus, we have the formula

\[ \int_D \{[V\,\partial_x]\,U+V[\partial_x U]\}\,dv =\int_S V\alpha U\,ds. \tag{5} \]

The differential operator

\[ \partial_{\bar x}=\frac{\partial}{\partial x_1} -i\frac{\partial}{\partial x_2} -j\frac{\partial}{\partial x_3} -k\frac{\partial}{\partial x_4} \tag{6} \]

will be called conjugate to the operator \(\partial_x\). Their product is

\[ \partial_x\partial_{\bar x}=\partial_{\bar x}\partial_x =\frac{\partial^2}{\partial x_1^2} +\frac{\partial^2}{\partial x_2^2} +\frac{\partial^2}{\partial x_3^2} +\frac{\partial^2}{\partial x_4^2}. \tag{7} \]

In formula (5) take the function \(V(x,z)=\partial_{\bar z}\dfrac{1}{r^2(x,z)}\), where \(r^2(x,z)= (x_1-z_1)^2+(x_2-z_2)^2+(x_3-z_3)^2+(x_4-z_4)^2\) is the square of the distance between the points \(x\) and \(z\), and the point \(z\) lies inside the domain \(D\). For convergence of the integral over the domain we remove from the domain \(D\) a certain ball of radius \(\varepsilon\) with center at the point \(z\); the domain thus obtained is denoted by \(D-\varepsilon\).

From formula (7) and from the fact that the function \(1/r^2(x,z)\) is harmonic, it follows that \(\partial_x V=0\). Therefore formula (5) takes the form

\[ \int_{D-\varepsilon} \left[\partial_{\bar z}\frac{1}{r^2(x,z)}\right] [\partial_x U]\,dv = \int_S \left[\partial_{\bar z}\frac{1}{r^2(x,z)}\right] \alpha U\,ds - \]

\[ - \int_{r(x,z)=\varepsilon} \left[\partial_{\bar z}\frac{1}{r^2(x,z)}\right] \alpha U\,ds = I_S+I_\varepsilon. \tag{8} \]

Let us pass in this formula to the limit as \(\varepsilon\to0\). The limit on the left-hand side of (8) exists, since the order of the singularity of the integrand, equal to three, ensures absolute convergence of the integral. Further:

\[ \lim I_\varepsilon =-\lim 2 \int_{r=\varepsilon} \frac{(x_1-z_1)-i(x_2-z_2)-j(x_3-z_3)-k(x_4-z_4)} {r^4(x,z)} \]

\[ \frac{(x_1-z_1)+i(x_2-z_2)+j(x_3-z_3)+k(x_4-z_4)} {r(x,z)} U(x)\,ds = -2\lim \int_{r=\varepsilon} \frac{U\,ds}{r^3(x,z)} = 2S_4U(z); \]

\(S_4\) is the surface area of the unit sphere \(x_1^2+x_2^2+x_3^2+x_4^2=1\).

Thus, as a result we obtain the representation for \(U(z)\):

\[ U(z)= \frac{1}{2S_4} \int_S \left[\partial_{\bar z}\frac{1}{r^2(x,z)}\right]\alpha U\,ds - \frac{1}{2S_4} \int_D \left[\partial_{\bar z}\frac{1}{r^2(x,z)}\right][\partial_x U]\,dv. \tag{9} \]

In the same way, formula (9) can be obtained for domains with piecewise smooth boundary, i.e. having a finite number of edges. Thus, we have:

Theorem 1. Let \(U(z)\) satisfy equation (2); then for it we have

\[ U(z)= \frac{1}{2S_4} \int_S \left[\partial_{\bar z}\frac{1}{r^2(x,z)}\right]\alpha U\,ds, \qquad z\in D. \tag{10} \]

If \(z\) does not belong to the domain \(D\), then from (5) we obtain

\[ 0=\frac{1}{2S_4}\int_S\left[\partial_z\frac{1}{r^2(x,z)}\right]\alpha U\,ds,\qquad z\notin D. \tag{11} \]

Formulas (10)—(11) give us an analogue of the Cauchy integral for \(U(z)\) satisfying (2). The analogue of Cauchy’s integral theorem will be the following assertion.

Theorem 2. If \(U(z)\) satisfies equation (1), then for any piecewise-smooth closed surface \(S\),

\[ \int_S \alpha U\,ds=0. \]

The proof follows directly from formula (3). The converse assertion is also valid—an analogue of Morera’s theorem.

Theorem 3. Let the function \(U(z)\) be continuous in \(\overline D\) and, for every closed piecewise-smooth surface \(S\) lying in \(\overline D\),

\[ \int_S \alpha U\,ds=0; \]

then \(U(z)\) is continuously differentiable in \(\overline D\) and satisfies equation (2).

The proof is carried out analogously to work (1).

With the aid of Theorem 1 one obtains an analogue of Liouville’s theorem.

Theorem 4. Let the function \(U(z)\) be continuously differentiable in the whole space, satisfy equation (2), and be bounded in modulus by some constant; then it is constant in the whole space.

For the proof one must write formula (10) for a sphere of radius \(R\), then differentiate it with respect to the variable \(z_i\), and let \(R\) tend to infinity; we obtain \(\partial U(z)/\partial z_i=0,\ i=1,2,3,4\).

Let us now consider an analogue of the Cauchy-type integral

\[ V(r)=\frac{1}{2S_4}\int_S\left[\partial_z\frac{1}{r^2(x,z)}\right]\alpha\varphi(x)\,ds, \tag{12} \]

where \(\varphi(x)\) is some quaternion-valued function given on \(S\). It is easy to verify that \(V(z)\) satisfies equation (2). If we additionally require of \(\varphi(x)\) Hölder continuity with some exponent \(\nu\), and require the surface to be a Lyapunov surface, then one can find the limiting values when the point \(z\) tends to some boundary point \(x_0\) from inside and from outside \(S\). The domain contained inside \(S\) will be denoted by \(D^+\), and the exterior of \(S\) by \(D^-\), and the corresponding limiting values by \(V^+(x_0)\) and \(V^-(x_0)\). These limiting values are the following:

\[ V^+(x_0)=\frac{\varphi(x_0)}{2}+ \frac{1}{2S_4}\int_S\left[\partial_{x_0}\frac{1}{r^2(x,x_0)}\right]\alpha\varphi\,ds, \]

\[ V^-(x_0)=-\frac{\varphi(x_0)}{2}+ \frac{1}{2S_4}\int_T\left[\partial_{z_0}\frac{0}{r^2(x,x_0)}\right]\alpha\varphi\,ds. \tag{13} \]

The derivation of these relations is carried out in the same way as in the case of the Cauchy integral for one complex variable. The integrals in (13) are understood in the sense of the Cauchy principal value.

From (13) follow the jump formulas on the boundary \(S\) for \(V(z)\):

\[ V^+(x_0)-V^-(x_0)=\varphi(x_0),\qquad V^+(x_0)+V^-(x_0)= \frac{1}{S_4}\int_S\left[\partial_{x_0}\frac{1}{r^2(x,x_0)}\right]\alpha\varphi\,ds. \tag{14} \]

In what follows we shall call continuously differentiable functions in \(D^+\) and \(D^-\), satisfying there equation (2), piecewise-holomorphic functions belonging to equation (2). Relations (14) make it possible to solve the following boundary p-

the problem of linear conjugation: to find a piecewise-holomorphic function belonging to equation (2), satisfying on \(S\) the condition

\[ V^{-}(x_0)=V^{+}(x_0)G+\varphi(x_0) \tag{15} \]

and vanishing at infinity. \(G\) is a certain constant quaternion distinct from zero. Indeed, the piecewise-holomorphic function equal to \(V(z)\) in \(D^{-}\) and to \(V(z)G\) in \(D^{+}\) satisfies equation (2) and is written by means of a Cauchy-type integral in view of formulas (14).

Boundary-value problems for system (1) in the domain \(D^{+}\), when two linear relations between \(u_1,u_2,u_3,u_4\) are prescribed on the boundary \(S\), were considered by M. Z. Solomyak \({}^{2}\); he showed that for the given system there are no normally solvable problems of this type.

With the aid of formulas (10)—(11) one can obtain a representation of an analytic function of two complex variables inside the domain \(D\) through its boundary values on \(S\).

Let \(w(\tilde z_1,\tilde z_2)=u_1+iu_2\) be an analytic function of two complex variables \(\tilde z_1=x_1+ix_2\) and \(\tilde z_2=x_3+ix_4\); then the Cauchy—Riemann conditions for it, in quaternionic notation, have the form

\[ \partial_{1x}=\left(\frac{\partial}{\partial x_1}+i\frac{\partial}{\partial x_2}+j\frac{\partial}{\partial x_3}-k\frac{\partial}{\partial x_4}\right)(u_1+iu_2)=0. \tag{16} \]

Arguing analogously to the derivation of (10), (11), we obtain

\[ u_1+iu_2=\frac{1}{2S_4}\int_S \left[\left(\frac{\partial}{\partial z_1}-i\frac{\partial}{\partial z_2}-j\frac{\partial}{\partial z_3}+k\frac{\partial}{\partial z_4}\right) \frac{1}{r^2(x,z)}\right]\cdot \]

\[ \cdot(\alpha_1+i\alpha_2+j\alpha_3-k\alpha_4)(u_1+iu_2)\,ds,\qquad z\in D^{+}; \tag{17} \]

\[ 0=\frac{1}{2S_4}\int_S \left[\left(\frac{\partial}{\partial z_1}-i\frac{\partial}{\partial z_2}-j\frac{\partial}{\partial z_3}+k\frac{\partial}{\partial z_4}\right) \frac{1}{r^2(x,z)}\right]\cdot \]

\[ \cdot(\alpha_1+i\alpha_2+j\alpha_3-k\alpha_4)(U_1+iU_2)\,ds,\qquad z\in D^{-}. \tag{18} \]

Further, if \(\varphi(x)=\varphi_1(x)+i\varphi_2(x)\), where \(\varphi_1\) and \(\varphi_2\) are real functions prescribed on \(S\), then the Cauchy-type integral

\[ V(z)=v_1+iv_2+jv_3+kv_4= \frac{1}{2S_4}\int_S \left[\overline{\partial}_{1z}\frac{1}{r^2(x,z)}\right] (\alpha_1+i\alpha_2+j\alpha_3-k\alpha_4)(\varphi_1+i\varphi_2)\,ds \tag{19} \]

will represent an analytic function of two complex variables \(\tilde z_1,\tilde z_2\), if \(v_3\equiv v_4\equiv0\) in \(D^{+}\).

Thus one obtains the following conditions, necessary and sufficient in order that a certain function \(\varphi(x)\), prescribed on \(S\), be the boundary value of some analytic function inside \(D\):

\[ V(z)\equiv0\quad \text{for } z\in D^{-};\qquad v_3(z)\equiv v_4(z)\equiv0\quad \text{for } z\in D^{+}. \tag{20} \]

Mathematical Institute named after V. A. Steklov
Academy of Sciences of the USSR

Received
1 VII 1963

REFERENCES

\({}^{1}\) G. Moisil, N. Theodorescu, Mathematica, 5, 141 (1931). \({}^{2}\) M. Z. Solomyak, DAN, 150, No. 1, 48 (1963)