A. YANUSHAUSKAS

THE CAUCHY PROBLEM FOR THE LAPLACE EQUATION AND A MULTIPLICATION OPERATION FOR HARMONIC FUNCTIONS

(Presented by Academician M. A. Lavrent’ev, 22 V 1964)

Let \(C^n\) be the space of \(n\) independent complex variables, \(R^n\) the \(n\)-dimensional Euclidean space; \(B\) a bounded domain of holomorphy in the space \(C^2\) (a domain of holomorphy is understood to mean a domain in which there exists at least one holomorphic function that cannot be continued holomorphically from this domain).

In this note we consider the Cauchy problem for the Laplace equation

\[ \Delta f \equiv \partial^2 f/\partial x^2+\partial^2 f/\partial y^2+\partial^2 f/\partial z^2=0. \tag{1a} \]

The investigation of the Cauchy problem is carried out in a complex domain. The results obtained for the Cauchy problem are used to introduce a multiplication operation \(\circ\) for harmonic functions.

Make the following change of variables in (1a):

\[ \xi=x+iy,\qquad \eta=x-iy,\qquad \zeta=z. \tag{2} \]

In these variables equation (1a) is rewritten in the form

\[ 4\partial^2 f/\partial \xi\,\partial \eta+\partial^2 f/\partial \zeta^2=0. \tag{1} \]

Consider the following problem.

Problem K. Find a solution \(f(\xi,\eta,\zeta)\) of equation (1) satisfying the conditions

\[ f\big|_{\zeta=0}=u(\xi,\eta),\qquad \partial f/\partial \zeta\big|_{\zeta=0}=v(\xi,\eta), \tag{3} \]

where \(u\) and \(v\) are functions holomorphic in the domain of holomorphy \(B\).

Represent the function \(f(\xi,\eta,\zeta)\) as the sum

\[ f(\xi,\eta,\zeta)=g(\xi,\eta,\zeta)+h(\xi,\eta,\zeta), \]

where \(g\) and \(h\) are solutions of equations (1), satisfying the conditions:

\[ g\big|_{\zeta=0}=u,\qquad \partial g/\partial \zeta\big|_{\zeta=0}=0;\qquad h\big|_{\zeta=0}=0,\qquad \partial h/\partial \zeta\big|_{\zeta=0}=v. \]

As is known \((^1)\), the functions \(g\) and \(h\) are represented by the series

\[ g=\sum_{n=0}^{\infty}(-1)^n\frac{\zeta^{2n}}{(2n)!}\,4^n \frac{\partial^{2n}u}{\partial \xi^n \partial \eta^n}, \tag{4a} \]

\[ h=\sum_{n=0}^{\infty}(-1)^n\frac{\zeta^{2n+1}}{(2n+1)!}\,4^n \frac{\partial^{2n}v}{\partial \xi^n \partial \eta^n}. \tag{4b} \]

This representation is valid only in the domain of absolute and uniform convergence of the series. We investigate the domain of convergence.

Lemma 1. If \(u\) and \(v\) are holomorphic in the bicylinder
\(D_0:\{|\xi-\xi_0|<r,\ |\eta-\eta_0|<r\}\), then the series (4a) and (4b) converge absolutely and uniformly in the circle
\(K_0:\{\xi=\xi_0,\ \eta=\eta_0,\ |\zeta|<r\}\).

Proof. We prove the assertion of the lemma for the function \(g\). For functions holomorphic in the bicylinder \(D_0\) the estimate \((^2)\)

\[ \left| \partial^{2n}u/\partial \xi^n \partial \eta^n \right|_{\xi=\xi_0,\ \eta=\eta_0} \leq M (n!)^2/r^{2n}, \tag{5} \]

is known, where

\[ M=\max_{D_0}|u(\xi,\eta)|. \]

Using this estimate, we obtain

\[ |g|\leq M\sum_{n=0}^{\infty}\frac{4^n(n!)^2}{(2n)!} \left|\frac{\zeta}{r}\right|^{2n}. \]

Since

\[ \lim_{n\to\infty} \frac{4^{n+1}[(n+1)!]^2}{(2n+2)!} : \frac{4^n(n!)^2}{(2n)!} =1, \]

the series

(4a) converges absolutely and uniformly in the disk \(K_0\). The assertion of the lemma for the function \(h\) is proved analogously.

Let the point \((\xi_0,\eta_0)\) belong to the domain of holomorphy \(B\). Consider the maximal bicylinder \(D:\{|\xi-\xi_0|<r,\ |\eta-\eta_0|<r\}\) contained in \(B\). By the lemma, the series (4) converge absolutely and uniformly on the set \(U(B):\{|\zeta|<r,\ (\xi_0,\eta_0)\in B\}\). The set \(U(B)\) contains an open set \(V\) in \(C^3\) such that \(B\subset V\subset U(B)\) \((^3)\).

We now take as the domain \(B\) the bicylinder \(D=D_1\times D_2\), where \(D_1\) is a domain in the \(\xi\)-plane with smooth boundary \(\Gamma_1\), and \(D_2\) is a domain in the \(\eta\)-plane with smooth boundary \(\Gamma_2\). The set \(\Gamma=\Gamma_1\times\Gamma_2\) is called the skeleton of the boundary of the bicylinder \(D\). As is known \((^2)\), any function \(u(\xi,\eta)\), holomorphic in \(D\) and Hölder-continuous in the closed domain \(\overline D\), can be represented in the following form:
\[ u(\xi,\eta)=-\frac{1}{4\pi^2}\iint_{\Gamma_1\Gamma_2} \frac{u(t,\tau)}{(t-\xi)(\tau-\eta)}\,d\tau\,dt. \tag{6} \]

Substituting (6) into the series (4a) and changing the order of summation and integration, we obtain
\[ g(\xi,\eta,\zeta)= -\frac{1}{4\pi^2}\iint_{\Gamma_1\Gamma_2} \frac{1}{(t-\xi)(\tau-\eta)} F\left(1,1;\frac12;-\frac{\zeta^2}{(t-\xi)(\tau-\eta)}\right) u(t,\tau)\,d\tau\,dt, \]
where \(F(a,\beta;\gamma;\lambda)\) is Gauss’s hypergeometric function. Analogously we obtain
\[ h(\xi,\eta,\zeta)= -\frac{1}{4\pi^2}\iint_{\Gamma_1\Gamma_2} \frac{\zeta}{(t-\xi)(\tau-\eta)} F\left(1,1;\frac32;-\frac{\zeta^2}{(t-\xi)(\tau-\eta)}\right) v(t,\tau)\,d\tau\,dt. \]

Theorem 1. For every bicylinder \(D\subset C^2\) with a smooth skeleton of the boundary \(\Gamma\), there exists a domain of holomorphy \(H(D)\subset C^3\) such that the solution of the problem \(K\), holomorphic in \(H(D)\), for arbitrary initial data holomorphic in \(D\), is represented in \(H(D)\) by the formula
\[ f(\xi,\eta,\zeta)= -\frac{1}{4\pi^2}\iint_{\Gamma_1\Gamma_2} \frac{1}{(t-\xi)(\tau-\eta)} \left[ F\left(1,1;\frac12;-\frac{\zeta^2}{(t-\xi)(\tau-\eta)}\right)u(t,\tau)+ \right. \]
\[ \left. +\zeta F\left(1,1;\frac32;-\frac{\zeta^2}{(t-\xi)(\tau-\eta)}\right)v(t,\tau) \right]d\tau\,dt. \tag{7} \]

For any point \(X\) on the boundary of the domain \(H(D)\) there exists a solution of problem \(K\) with initial data holomorphic in \(D\), having a singularity at the point \(X\).

Proof. As is known \((^4)\), the hypergeometric function \(F(1,1;\gamma;\lambda)\) is single-valued and holomorphic in the \(\lambda\)-plane cut along the ray \([1,\infty)\), and at the point \(\lambda=1\) it has a singularity of order \(2-\gamma\). \(f(\xi,\eta,\zeta)\) in (7) is holomorphic everywhere where
\[ F\left(1,1;\frac12;-\frac{\zeta^2}{(t-\xi)(\tau-\eta)}\right) \]
and
\[ F\left(1,1;\frac32;-\frac{\zeta^2}{(t-\xi)(\tau-\eta)}\right) \]
are holomorphic. Let \(\Omega\) be the union of all surfaces
\[ Q:\{\zeta^2+(t-\xi)(\tau-\eta)=0,\ (t,\tau)\in\Gamma\}. \]
Consider the connected component \(E\) of the complement \(C\Omega\) containing an open neighborhood \(V\subset U(D)\) of the set \(D\) in \(C^3\). This component will be the domain \(H(D)\). We shall show that \(H(D)\) is a domain of holomorphy. \(H(D)\) is the intersection of domains of holomorphy and contains an open set \(V\subset U(B)\) in \(C^3\); by Theorem 11.7 of \((^2)\), \(H(D)\) is a domain of holomorphy. The second assertion of the theorem is obvious.

Theorem 2. The set of harmonic functions \(u\) regular in the ball
\[ (\operatorname{Re}x)^2+(\operatorname{Re}y)^2+(\operatorname{Re}z)^2<R^2 \]
and the set of all possible pairs \((u,v)\) of functions holomorphic in the domain
\[ D:\{|x+iy|<R,\ |x-iy|<R\} \]
are in one-to-one correspondence.

Proof. Formula (7) in this case takes the form

\[ \begin{aligned} f(\xi,\eta,\zeta)= -\frac{1}{4\pi^2}\int_0^{2\pi}\int_0^{2\pi} \frac{1}{(e^{i\varphi}-\xi)(e^{i\psi}-\eta)} \Bigg[ &F\left(1,1;\frac12;-\frac{\zeta^2}{(e^{i\varphi}-\xi)(e^{i\psi}-\eta)}\right) du(e^{i\varphi},e^{i\psi}) \\ &+\zeta F\left(1,1;\frac32;-\frac{\zeta^2}{(e^{i\varphi}-\xi)(e^{i\psi}-\eta)}\right) v(e^{i\varphi},e^{i\psi}) \Bigg]\,d\psi\,d\varphi . \end{aligned} \tag{7a} \]

The singularities of the functions
\(F(1,1;\gamma;-\zeta^2/(e^{i\varphi}-\xi)(e^{i\psi}-\eta))\)
\((\gamma=1/2,\,3/2)\) are situated on the surfaces
\(\zeta^2+(e^{i\varphi}-\xi)(e^{i\psi}-\eta)=0\).
Putting \(\operatorname{Im}x=\operatorname{Im}y=\operatorname{Im}z=0\), we obtain
\((\operatorname{Re}x)^2+(\operatorname{Re}y)^2+(\operatorname{Re}z)^2 +R^2\cos(\varphi+\psi)-R(\operatorname{Re}x)(\cos\varphi+\cos\psi) +R(\operatorname{Re}y)(\sin\varphi-\sin\psi)=0\),
\(R^2\sin(\varphi+\psi)-R(\operatorname{Re}x)(\sin\varphi+\sin\psi) +R(\operatorname{Re}y)(\cos\psi-\cos\varphi)=0\).

From these equalities we obtain
\((\operatorname{Re}x)^2+(\operatorname{Re}y)^2+(\operatorname{Re}z)^2=R^2\).
It follows that the function \(f(\xi,\eta,\zeta)\) is regular in the ball

\[ (\operatorname{Re}x)^2+(\operatorname{Re}y)^2+(\operatorname{Re}z)^2<R^2 . \tag{8} \]

Let us show conversely that if the harmonic function \(f\) is regular in the ball (8), then \(u\) and \(v\) are holomorphic in \(D\). Consider two particular cases of problem K:

\[ g_\alpha^\beta\big|_{\zeta=0}=\xi^\alpha\eta^\beta,\qquad \partial g_\alpha^\beta/\partial\zeta\big|_{\zeta=0}=0;\qquad h_\alpha^\beta\big|_{\zeta=0}=0,\qquad \partial h_\alpha^\beta/\partial\zeta\big|_{\zeta=0}=\xi^\alpha\eta^\beta . \]

The solutions of these problems have the form

\[ g_\alpha^\beta=\xi^\alpha\eta^\beta F\left(-\alpha,-\beta;\frac12;-\zeta^2/\xi\eta\right), \qquad h_\alpha^\beta=\zeta\xi^\alpha\eta^\beta F\left(-\alpha,-\beta;\frac32;-\zeta^2/\xi\eta\right). \]

For natural \(\alpha\) and \(\beta\), \(g_\alpha^\beta\) differs only by a constant factor from the spherical function
\(r^{2\nu}P_{2\nu}^{\mu}(\cos\theta)e^{i\mu\varphi}\), and \(h_\alpha^\beta\) differs only by a constant factor from
\(r^{2\nu+1}P_{2\nu+1}^{\mu}(\cos\theta)e^{i\mu\varphi}\), where
\(\nu=\max(\alpha,\beta)\), \(\mu=|\alpha-\beta|\),
and \(P_k^\rho(\omega)\) is the associated Legendre polynomial. As is known \((^5)\), every harmonic function regular in the ball (8) can be expanded into an absolutely and uniformly convergent series in spherical functions

\[ f(x,y,z)= \sum_{\nu=0}^{\infty}\sum_{\mu=0}^{\nu} r^{2\nu}e^{i\mu\varphi} \left[ a_\nu^\mu P_{2\nu}^{\mu}(\cos\theta) + r b_\nu^\mu P_{2\nu+1}^{\mu}(\cos\theta) \right]. \tag{9} \]

From (9), for \(z=0\) we have

\[ f\big|_{z=0}= \sum_{\nu=0}^{\infty}\sum_{\mu=0}^{\nu} A_\nu^\mu (x+iy)^\nu (x-iy)^{\nu+\mu}, \tag{10} \]

\[ \frac{\partial f}{\partial z}\bigg|_{z=0}= \sum_{\nu=0}^{\infty}\sum_{\mu=0}^{\nu} B_\nu^\mu (x+iy)^\nu (x-iy)^{\nu+\mu}. \]

The series (10) converge absolutely and uniformly in the circle
\(|x+iy|<R,\ |x-iy|<R,\ \xi=\eta\), and therefore they will converge for
\(|\xi|<R,\ |\eta|<R\), i.e. in the domain
\(D:\{|\xi|<R,\ |\eta|<R\}\). The theorem is proved.

Consider the set of harmonic functions \(g\), regular in the ball (8) and satisfying the conditions

\[ g\big|_{\zeta=0}=u(\xi,\eta),\qquad \partial g/\partial\zeta\big|_{\zeta=0}=0 . \tag{11} \]

By Theorem 2, these harmonic functions are in one-to-one correspondence with the functions \(u(\xi,\eta)\) holomorphic in the domain
\(D:\{|\xi|<R,\ |\eta|<R\}\). Denote by \(F(D)\) the ring of functions holomorphic in the domain \(D\), and by \(G(F)\) the set of harmonic functions satisfying the conditions (11). On the set \(G(F)\) one can introduce a ring structure as follows: let
\(g_1\big|_{\zeta=0}=u_1,\ g_2\big|_{\zeta=0}=u_2\); then
\(g_3=g_1+g_2\) satisfies the conditions
\(g_3\big|_{\zeta=0}=u_1+u_2,\ \partial g_3/\partial\zeta\big|_{\zeta=0}=0\),
and the multiplication operation \(\circ\) is defined as follows:
\(g_4=g_1\circ g_2\), if
\(g_4\big|_{\zeta=0}=u_1\cdot u_2,\ \partial g_4/\partial\zeta\big|_{\zeta=0}=0\).
Endowed with these operations, \(G(F)\) is a commutative ring without zero divisors. The identity in \(G(F)\) is the function \(g\equiv 1\).

We denote by \(H(F)\) the set of all harmonic functions regular in the ball (8). On \(H(F)\) one can define the structure of a \(G(F)\)-module as follows: let \(f\in H(F)\) and let it satisfy the conditions
\(f|_{\zeta=0}=u\), \(\partial f/\partial \zeta|_{\zeta=0}=v\); then \(f_1=g\circ f\) satisfies the conditions
\(f_1|_{\zeta=0}=u\cdot u_1\), \(\partial f/\partial \zeta|_{\zeta=0}=v u_1\), where \(u_1=g|_{\zeta=0}\). In the ring \(G(F)\) one can introduce a norm in the following way: if \(g|_{\zeta=0}=u_1\), \(g\in G(F)\), then set \(\|g\|=\max u_1\). In the \(G(F)\)-module \(H(F)\) one can also introduce a norm in the following way: let \(f\in H(F)\); then \(f=g+h\), where \(g\in G(F)\) and \(\partial h/\partial\zeta\in G(F)\); set
\(\|f\|=\sqrt{\|g\|^2+\|\partial h/\partial\zeta\|^2}\). It is obvious that the norm thus introduced is consistent with all operations defined on \(G(F)\) and on \(H(F)\), i.e., these operations are continuous in this norm.

As is known \((^2)\), every function holomorphic in the bicylinder \(D:\{|\xi|<R,\ |\eta|<R\}\) can be represented by a power series absolutely and uniformly convergent in \(D\). Denote by \(F_0(D)\) the subring of the ring \(F(D)\) consisting of all functions holomorphic in \(D\) whose power series converge absolutely in the closed bicylinder \(\overline D\). Denote by \(H(F_0)\) the set of functions harmonic in the ball (8) for which the series in spherical functions converge absolutely in the closed ball. Denote by \(G(F_0)\) the subset of functions \(g\) from \(H(F_0)\) satisfying the condition
\(\partial g/\partial\zeta|_{\zeta=0}=0\).

Theorem 3. If \(u,v\in F_0(D)\), then the harmonic function \(f\) satisfying the conditions \(f|_{\zeta=0}=u\), \(\partial f/\partial\zeta|_{\zeta=0}=v\), belongs to \(H(F_0)\), and conversely.

Proof. Let
\(u=\xi^m\eta^n\), \(D_\varepsilon:\{|\xi|<R+\varepsilon,\ |\eta|<R+\varepsilon\}\). Applying formula (7) to \(u\) and \(D_\varepsilon\), and estimating the integral in (7), we obtain

\[ \max_{H(D)} |g_m^n|\le K\max_{\overline D} |\xi^m\eta^n|, \]

where

\[ K=\max_{\overline{H(D)}}\left| \frac{1}{[(R+\varepsilon)e^{i\varphi}-\xi][(R+\varepsilon)e^{i\psi}-\eta]} \times \right. \]

\[ \left. \times F\left(1,1;\frac12;- \frac{\zeta^2}{[(R+\varepsilon)e^{i\varphi}-\xi][(R+\varepsilon)e^{i\psi}-\eta]} \right) \right|. \]

An analogous estimate can be obtained for the harmonic function \(h_m^n\) satisfying the conditions
\(h_m^n|_{\zeta=0}=0\), \(\partial h_m^n/\partial\zeta|_{\zeta=0}=\xi^m\eta^n\). From these estimates the direct assertion of the theorem follows.

Let us prove the converse assertion of the theorem. Suppose we have

\[ f(\xi,\eta,\zeta)= \sum_{n,m=0}^{\infty} a_{mn}\xi^m\eta^n F\left(-m,-n;\frac12;-\frac{\zeta^2}{\xi\eta}\right)+ \]

\[ +\zeta \sum_{n,m=0}^{\infty} b_{nm}\xi^m\eta^n F\left(-m,-n;\frac32;-\frac{\zeta^2}{\xi\eta}\right). \tag{12} \]

By hypothesis, the series (12) converge absolutely for \(\zeta=0\), \(\xi=\eta\), \(|\xi|=R\). By Abel’s theorem \((^2)\) these series converge absolutely for all \(|\xi|\le R\), \(|\eta|=R\).

Corollary. The normed rings \(F_0(D)\) and \(G(F_0)\) are isomorphic.

Institute of Mathematics
Siberian Branch of the Academy of Sciences of the USSR

Received
29 IV 1964

CITED LITERATURE

1. A. V. Bitsadze, Trudy Tbilisi State University, 84 (1961).
2. B. A. Fuks, Introduction to the Theory of Analytic Functions of Several Complex Variables, Moscow, 1962.
3. J. Leray, Collection of papers. Mathematics, 3, 5 (1959).
4. A. Kratzer, W. Franz, Transcendental Functions, Moscow, 1963.
5. R. Courant, D. Hilbert, Methods of Mathematical Physics, Moscow, 1945.