UDC 517.53 : 517.947.42

MATHEMATICS

Corresponding Member of the Academy of Sciences of the USSR A. V. BITSADZE

ON A CRITERION FOR THE CONVERGENCE OF THE GRADIENTS OF A SEQUENCE OF HARMONIC FUNCTIONS

In the classical theory of functions, an important role is played by the following theorem on the convergence of a sequence \(\{f_n(z)\}\), \(n=1,2,\ldots\), of functions holomorphic in a domain \(D\) of the plane of the complex variable \(z=x+iy\): if the sequence \(\{u_n(x,y)=\operatorname{Re} f_n(z)\}\) converges to zero uniformly in the domain \(D\), and the sequence \(\{v_n(x,y)=\operatorname{Im} f_n(z)\}\) converges to zero at a fixed point \(z_0 \in D\), then \(\{f_n(z)\}\) converges to zero uniformly in every bounded closed domain \(D^*\) belonging to the domain \(D\).

The present note is devoted to establishing a multidimensional analogue of this theorem.

For simplicity of notation, below we shall restrict ourselves to consideration of the three-dimensional case.

If a sequence \(\{u_n(x,y,z)\}\), \(n=1,2,\ldots\), of harmonic functions regular in a domain \(D\) of the space of the variables \(x,y,z\) has the properties: a) the sequence \(\{\partial u_n/\partial x\}\) converges to zero in the domain \(D\) uniformly with respect to the variables \(x,y,z\); b) the sequence \(\{\partial u_n/\partial y\}\) converges to zero in the domain \(D\) uniformly with respect to the variables \(y,z\), and c) the sequence \(\{\partial u_n/\partial z\}\) converges to zero at a fixed point of the domain \(D\), for example, at the point \((0,0,0)\), then the sequence \(\{\operatorname{grad} u_n(x,y,z)\}\) converges to zero uniformly with respect to \(x,y,z\) in every bounded closed domain \(D^*\) lying in the domain \(D\).

For the purpose of proving our assertion, let us note that it is always possible to indicate a positive number \(r\) such that the closed ball \(C(r;x_0,y_0,z_0)\) of radius \(r\) with center at any point \((x_0,y_0,z_0)\in D^*\) will lie in the domain \(D\).

Inside the ball \(C(r;x_0,y_0,z_0)\), for each harmonic function of the sequence \(\{u_n(x,y,z)\}\) the integral representation \((^1,^2)\) holds

\[ u_n(x,y,z)=\frac{1}{4\pi r^2}\iint_S \left[ \frac{(r^2-\xi^2-\eta^2-\zeta^2)(x-\xi)}{\Delta R^{1/2}} +\frac{x+\xi}{R^{1/2}} -\operatorname{Ar\,sh}\frac{x-\xi}{\Delta^{1/2}} \right]\times \]

\[ \times \frac{\partial u_n(\xi,\eta,\zeta)}{\partial \xi}\,dS +\gamma_n(y,z;x_0,y_0,z_0), \tag{1} \]

where \(S\) is the sphere
\[ (\xi-x_0)^2+(\eta-y_0)^2+(\zeta-z_0)^2=r^2, \]
\[ \Delta=(y-\eta)^2+(z-\zeta)^2,\qquad R=(x-\xi)^2+\Delta, \]
and \(\gamma_n(y,z;x_0,y_0,z_0)\) is a completely determined regular harmonic function of the variables \(y,z\) in the cylinder
\[ (y-y_0)^2+(z-z_0)^2<r^2. \]

Denote by \(\delta\) and \(\delta_1\) positive numbers satisfying the conditions \(\delta<\delta_1<r\).

Computing the partial derivatives \(\partial u_n/\partial y\) and \(\partial u_n/\partial z\) from formula (1) at the points \((0,y,z)\) and \((0,0,0)\), respectively, by virtue of conditions a), b), and c), we conclude that the sequence \(\{\partial\gamma_n(y,z;0,0,0)/\partial y\}\) converges to zero uniformly for \(y^2+z^2\le \delta_1^2\), and the sequence \(\{\partial\gamma_n(y,z;0,0,0)/\partial z\}\) converges to zero for \(y=z=0\). Hence, by virtue of the theorem formulated at the beginning of the present note, there follows uniform convergence to zero

of the sequence \(\{\operatorname{grad}\gamma_n(y,z;0,0,0)\}\) for \(y^2+z^2\leq \delta^2\). Taking this circumstance into account, on the basis of conditions a), b), and c), again from formula (1) we conclude that the sequence \(\{\operatorname{grad}u_n(x,y,z)\}\) converges uniformly to zero in the ball \(C(\delta;0,0,0)\).

Moving the center of the sphere \(C(r;0,0,0)\) to the point \((x_0,y_0,z_0)\) along a continuous path \(L\) lying in the domain \(D^*\), and taking into account that condition c) may be regarded as fulfilled at the point \((x_0,y_0,z_0)\), by repeating the argument just given we become convinced of the uniform convergence to zero of the sequence \(\{\operatorname{grad}u_n(x,y,z)\}\) in the sphere \(C(\delta;x_0,y_0,z_0)\). Hence, by virtue of the Heine–Borel lemma, the validity of our assertion follows immediately.

The assertion proved is, in an obvious way, rephrased for a sequence of vectors \(\{P_n(u_n,v_n,w_n)\}\) that are regular solutions of the system \(\operatorname{div} P_n=0,\ \operatorname{rot} P_n=0\).

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

Received
12 III 1966

REFERENCES

1. A. V. Bitsadze, DAN, 159, No. 5 (1965).
2. A. V. Bitsadze, Boundary Value Problems for Elliptic Equations of the Second Order, “Nauka,” 1966.