UDC 517.53

MATHEMATICS

I. S. ARSHON, M. A. PAK

AN EXAMPLE OF A HARMONIC FUNCTION BOUNDED OUTSIDE A BODY OF REVOLUTION AND GROWING INSIDE IT

(Presented by Academician I. N. Vekua, January 23, 1967)

The present note is devoted to the following problem, discussed in papers \((^{1-6})\). Let \(D\) be some unbounded domain in three-dimensional space \((x,x_1,x_2)\), and let \(u(P)\) be a function harmonic in this domain and continuous, together with its gradient, up to the boundary \(S\) of the domain \(D\). It is required to estimate the upper bound \(\varphi_D(P)\) of functions \(\varphi(P)\) such that from the conditions

\[ |u(P)|<C\exp\{\varphi(P)\},\qquad P\in D, \]

\[ |u(P)|+|\operatorname{grad}u(P)|<C,\qquad P\in S, \]

there follows the boundedness of \(u(P)\) everywhere inside \(D\).

We shall call the function \(\varphi_D(P)\) the growth indicator for the domain \(D\).

In the paper of M. A. Evgrafov \((^6)\), a lower estimate for the growth indicator is found in the case of an arbitrary domain whose boundary satisfies certain smoothness conditions. In the note \((^4)\), an example is constructed of a harmonic function in the whole space, bounded outside a circular half-cylinder and unbounded inside. The growth of this function gives an upper estimate for the growth indicator for every domain containing within itself a circular half-cylinder. Naturally, this estimate gives a satisfactory result only for domains which themselves are close to a half-cylinder.

In the present note, by somewhat generalizing the construction proposed in \((^4)\), we obtain an upper estimate for the growth indicator sufficiently sharp for a broader class of domains.

Theorem 1. Let the domain \(V\) be a body of revolution with generatrix

\[ x_1=\tfrac12 h(x);\qquad x>x_0;\qquad h(x)\ge r>0. \]

Suppose that the function \(h(x)\) satisfies the following conditions:

1. \(h(z)\) is regular and has no zeros in the domain \(G_{x_0}\):

\[ |\operatorname{Im} z|<\tfrac12 h(\operatorname{Re} z),\qquad \operatorname{Re} z>x_0. \]

1. In the indicated domain,

\[ \lim_{|z|\to\infty} h'(z)=0. \]

Then there exists a function \(u_\eta(x,x_1,x_2)\), harmonic in the whole space, bounded outside the body of revolution \(V\), and unbounded inside, and moreover

\[ |u_\eta(x,x_1,x_2)|<C\exp\{\varphi_{2\eta}(x)\};\qquad (x,x_1,x_2)\in V, \]

where

\[ \varphi_\eta(x)=\exp\left\{(\pi+\eta)\int_{x_0}^{x}\frac{dt}{h(t)}\right\}, \]

and \(\eta>0\) is arbitrarily small.

Proof. Put

\[ \varphi_\eta(z)=\exp\left\{(\pi+\eta)\int_{x_0}^{z}\frac{dt}{h(t)}\right\}. \]

We shall further denote by \(G_{\alpha,x}\) the domain of the complex \(z\)-plane:

\[ |\operatorname{Im} z|<\frac{\alpha}{2}h(\operatorname{Re}z),\qquad \operatorname{Re}z>x, \]

and by \(L_{\alpha,x}\) the contour bounding this domain. Finally, let

\[ u_\eta(x,x_1,x_2)=\frac{1}{2\pi i}\int_{L_{\alpha,x_0+1}} \frac{\exp\{\varphi_\eta(z)\}}{\sqrt{(z-x)^2+\rho^2}}\,dz;\qquad \rho=\sqrt{x_1^2+x_2^2},\quad x<x_0, \tag{1} \]

where the regular branch of the root is fixed by the condition

\[ \operatorname{Re}\sqrt{(z-x)^2+\rho^2}>0, \]

and \(\alpha<1\) is chosen so that the inequality

\[ \pi>\frac{\pi+\eta}{2}\alpha>\frac{\pi}{2} \]

is satisfied.

From the conditions imposed on \(h(z)\) there easily follows the estimate

\[ \left|\exp\left\{\varphi_\eta\left(x\pm i\frac{\alpha}{2}h(x)\right)\right\}\right| = \exp\left\{\varphi_\eta(x)\cos\left[\frac{\pi+\eta}{2}\alpha(1+o(1))\right]\right\}, \qquad x\to\infty . \]

It is also clear that, as \(x\to\infty\), the function \(\varphi_\eta(x)\) grows faster than any power of \(x\). Therefore the integral (1) converges absolutely. Differentiating under the integral sign, we are convinced that \(u_\eta(x,x_1,x_2)\) is a harmonic function in the half-space \(x<x_0\); replacing in (1) the contour of integration \(L_{\alpha,x_0+1}\) by \(L_{\alpha,x+1}\), we carry out a harmonic continuation of this function to the entire space \((x,x_1,x_2)\).

Let us show that \(u_\eta(x,x_1,x_2)\) is bounded outside the body \(V\). Indeed, in this case either \(x<x_0\), or \(\rho\ge \tfrac12 h(x)\). Hence the branch points of the root \(z=x\pm i\rho\) lie outside the domain \(G_{x_0}\). But \(G_{x_0}\) contains within itself the contour \(L_{\alpha,x_0+1}\), since \(\alpha<1\). Therefore

\[ u_\eta(x,x_1,x_2)=\frac{1}{2\pi i}\int_{L_{\alpha,x_0+1}} \frac{\exp\{\varphi_\eta(z)\}}{\sqrt{(z-x)^2+\rho^2}}\,dz;\quad \rho=\sqrt{x_1^2+x_2^2};\quad (x,x_1,x_2)\notin V . \tag{2} \]

Since

\[ \left|(z-x)^2+\rho^2\right| = |z-(x+i\rho)|\cdot|z-(x-i\rho)|, \]

where \(z\in L_{\alpha,x_0+1}\), while \(x\pm i\rho\notin G_{x_0}\), we have

\[ \left|\sqrt{(z-x)^2+\rho^2}\right|\ge d>0, \]

where \(d\) is the distance between \(L_{\alpha,x_0+1}\) and \(L_{x_0}\). According to (2), this gives

\[ |u_\eta(x,x_1,x_2)|\le \frac{1}{2\pi d} \int_{L_{\alpha,x_0+1}} |\exp\{\varphi_\eta(z)\}|\,|dz| \le \mathrm{const}. \]

Thus, the boundedness of \(u_\eta(x,x_1,x_2)\) outside \(V\) is proved.

Now let us estimate from above the growth of \(u_\eta(x,x_1,x_2)\) inside the body \(V\). Obviously,

\[ |u_\eta(x,x_1,x_2)|\le \left| \int_{x+1-\frac{i\alpha}{2}h(x+1)}^{x+1+\frac{i\alpha}{2}h(x+1)} \exp\{\varphi_\eta(z)\}\,dz \right| +\mathrm{const} \le \]

\[ \le Ch(x+1)\exp\{\varphi_\eta(x+1)\}; \]

therefore

\[ |u_\eta(x,x_1,x_2)|<C\exp\{\varphi_{2\eta}(x)\}. \]

It remains to establish that the function \(u_\eta(x,x_1,x_2)\) constructed by us is not bounded inside \(V\). Putting \(\rho=0\), we have

\[ u_\eta(x,0,0)=\frac{1}{2\pi i} \int_{L_{\alpha,x+1}} \frac{\exp\{\varphi_\eta(z)\}}{z-x}\,dz = \exp\{\varphi_\eta(x)\} +\frac{1}{2\pi i} \int_{L_{\alpha,x_0+1}} \frac{\exp\{\varphi_\eta(z)\}}{z-x}\,dz . \]

The integral is bounded, while \(\exp\{\varphi_n(x)\}\) grows as \(x \to \infty\). The theorem is proved.

From the result obtained it follows that, for any domain \(D\) containing within itself the body of revolution \(V\) specified in the statement, we have

\[ \varphi_D(P) \leqslant \exp\left\{(\pi+\varepsilon)\int_{x_0}^{x}\frac{dt}{h(t)}\right\} \]

with arbitrarily small \(\varepsilon>0\).

To judge the sharpness of the estimate obtained, we give one assertion which follows easily from the results of M. A. Evgrafov\({}^{6}\) and the known Phragmén—Lindelöf theorems for regular functions:

Theorem 2. Suppose that the function \(h(x)\), in addition to the conditions indicated above, also has the following property:

\[ \int_{x_0}^{\infty}\frac{h'^2(t)}{h(t)}\,dt<\infty, \]

and suppose that the domain \(D\) is entirely contained inside the cylinder with directrix \(x_1=\frac12 h(x)\) and generators parallel to the axis \(Ox_2\) (in particular, our body of revolution \(V\) may serve as the domain \(D\)). Then

\[ \varphi_D(P) \geqslant \exp\left\{(\pi-\varepsilon)\int_{x_0}^{x}\frac{dt}{h(t)}\right\} \]

with any \(\varepsilon>0\).

In conclusion, the authors of the present note express their gratitude to M. A. Evgrafov for his constant attention and interest in this work.

Moscow Institute
of Electronic Machine Building

Received
20 XII 1966

CITED LITERATURE

\({}^{1}\) M. A. Evgrafov, I. A. Chegis, DAN, 134, No. 2, 259 (1960).
\({}^{2}\) I. A. Chegis, DAN, 136, No. 3, 556 (1961).
\({}^{3}\) I. S. Arshon, M. A. Evgrafov, DAN, 142, No. 4, 762 (1962).
\({}^{4}\) I. S. Arshon, M. A. Evgrafov, DAN, 143, No. 1, 9 (1962).
\({}^{5}\) I. S. Arshon, M. A. Evgrafov, DAN, 147, No. 4, 755 (1962).
\({}^{6}\) M. A. Evgrafov, Izv. AN SSSR, ser. matem., 17, No. 4, 843 (1963).