Abstract
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MATHEMATICS
I. S. ARSHON and M. A. EVGRAFOV
ON THE GROWTH OF FUNCTIONS HARMONIC IN A CYLINDER AND BOUNDED ON ITS SURFACE TOGETHER WITH THE NORMAL DERIVATIVE
(Presented by Academician M. V. Keldysh on 16 IX 1961)
In the papers of M. A. Evgrafov and I. A. Chegis \((^{1,2})\), for harmonic functions of three variables, theorems were proved analogous to the Phragmén–Lindelöf theorem for analytic functions. In these works harmonic functions \(u(x,x_1,x_2)\) were considered in the half-cylinder
\[ 0<x<\infty,\qquad (x_1,x_2)\subset D, \tag{1} \]
where the domain \(D\) in paper \((^1)\) is a disk, and in paper \((^2)\) a rectangle. The results obtained in \((^{1,2})\) are as follows:
If on the lateral surface of the half-cylinder (1) \(u(x,x_1,x_2)=0\) and
\(\dfrac{\partial}{\partial n}u(x,x_1,x_2)\) is bounded, while inside it
\[ |u(x,x_1,x_2)|<M\exp\exp \frac{\pi}{h+\eta}x,\qquad \eta>0,\ x>0, \]
where \(h\) is, in the case of the disk, the diameter, and in the case of the rectangle, the smaller side, then \(u(x,x_1,x_2)\equiv 0\).
It was also shown there that the constants found are sharp. The proofs were based on a uniqueness theorem for Dirichlet series.
The purpose of the present note is to prove a similar result, but already for the case of the half-cylinder (1) with an arbitrary domain \(D\). (Concerning the domain \(D\), we shall assume only that the lateral surface of the half-cylinder (1) is a Lyapunov surface.)
The proof of the theorem given below is completely unrelated to Dirichlet series.
Theorem. If the function \(u(x,x_1,x_2)\), harmonic in the half-cylinder (1), satisfies on the boundary \(\Gamma\) of this half-cylinder the conditions
\[ |u(x,x_1,x_2)|+\left|\frac{\partial}{\partial n}u(x,x_1,x_2)\right| =O(x^{-\alpha}),\qquad 0<\alpha<1,\ x\to\infty, \tag{2} \]
and inside it the condition
\[ \iint_D |u(x,x_1,x_2)|\,dx_1dx_2 <M\exp\exp \frac{\pi}{h+\eta}x,\qquad \eta>0,\ x>0, \tag{3} \]
where \(h\) is the width of the minimal strip containing \(D\), then
\[ |u(x,x_1,x_2)|=O(x^{-\alpha}),\qquad x\to\infty, \]
uniformly in \((x_1,x_2)\subset D\).
In the proof we use two well-known facts:
Lemma 1. Let \(f(z)\) be an analytic function, regular in the half-plane \(\operatorname{Re} z \geqslant \sigma > 0\). If \(f(z)\) satisfies the inequalities
\[ \ln |f(z)| < M |z|, \qquad \operatorname{Re} z \geqslant \sigma > 0; \]
\[ |f(\sigma+i\tau)| < M e^{a_1\tau}, \qquad |f(\sigma-i\tau)| < M e^{a_2\tau}, \qquad \tau > 0; \]
\[ |f(z)| < M \exp\left[-\frac{\rho}{\pi} z \ln z\right], \qquad z \geqslant \sigma \]
and \(\rho > |a_2-a_1|\), then \(f(z)\equiv 0\).
Lemma 2. Let \(f(\xi_1,\xi_2)\) be an analytic function of two complex variables, regular in the domain
\(\operatorname{Re}\sqrt{\xi_1^2+\xi_2^2}\geqslant \sigma>0\). If
\[ f(z\cos\varphi,\ z\sin\varphi)=0 \]
for \(\varphi_0-\delta<\varphi<\varphi_0+\delta\) and \(\operatorname{Re} z\geqslant \sigma\), then
\(f(\xi_1,\xi_2)\equiv 0\).
Proof of the theorem. We shall obtain the assertion of the theorem by establishing the formula
\[ u(x,x_1,x_2)= \iint_{\Gamma}\frac{1}{r}\frac{\partial u}{\partial n}\,dS - \iint_{\Gamma}u\,\frac{\partial(1/r)}{\partial n}\,dS \tag{4} \]
(where \(\Gamma\) is the boundary of the half-cylinder (1), \(dS\) is the area element), expressing the value of \(u(x,x_1,x_2)\) at any interior point of the half-cylinder in terms of the values of \(u\) and \(\partial u/\partial n\) on its boundary. The estimate of the integral in formula (4) is easily obtained by the usual methods of investigating single- and double-layer potentials (see, for example, (3)).
To establish formula (4), we carry out the following arguments. Consider the Fourier transform of our harmonic function with respect to the variables \(x_1,x_2\):
\[ \widetilde u(x,\xi_1,\xi_2)= \iint_D u(x,x_1,x_2)\exp[i x_1\xi_1+i x_2\xi_2]\,dx_1\,dx_2 . \]
Since \(u(x,x_1,x_2)\) satisfies Laplace’s equation, simple transformations give us, for \(\widetilde u(x,\xi_1,\xi_2)\), the equation
\[ \frac{\partial^2 \widetilde u}{\partial x^2} = (\xi_1^2+\xi_2^2)\widetilde u+\psi(x,\xi_1,\xi_2), \tag{5} \]
where
\[ \psi(x,\xi_1,\xi_2) = -\int_{\gamma} \exp[i x_1\xi_1+i x_2\xi_2]\, u(x,x_1,x_2)(\xi_2\,dx_1-\xi_1\,dx_2) + \]
\[ +\int_{\gamma} \exp[i x_1\xi_1+i x_2\xi_2] \left( \frac{\partial u}{\partial x_2}\,dx_1 - \frac{\partial u}{\partial x_1}\,dx_2 \right); \tag{6} \]
here \(\gamma\) is the boundary of the domain \(D\).
Solving equation (5), we find for \(\widetilde u(x,\xi_1,\xi_2)\) the formula
\[ \widetilde u(x,\xi_1,\xi_2) = \Phi(x,\xi_1,\xi_2) + f(\xi_1,\xi_2)\exp\left[x\sqrt{\xi_1^2+\xi_2^2}\right], \tag{7} \]
where
\[ \Phi(x,\xi_1,\xi_2) = \left\{ \frac{1}{2}\widetilde u(0,\xi_1,\xi_2) - \frac{\widetilde u'_x(0,\xi_1,\xi_2)} {2\sqrt{\xi_1^2+\xi_2^2}} \right\} \exp\left[-x\sqrt{\xi_1^2+\xi_2^2}\right] - \]
\[ - \int_0^\infty \frac{ \exp\left[-\sqrt{\xi_1^2+\xi_2^2}\,|x-t|\right] } {2\sqrt{\xi_1^2+\xi_2^2}} \, \psi(t,\xi_1,\xi_2)\,dt, \tag{8} \]
where
\[ f(\xi_1,\xi_2)=\frac12\tilde u(0,\xi_1,\xi_2)+ \frac{\tilde u'_x(0,\xi_1,\xi_2)}{2\sqrt{\xi_1^2+\xi_2^2}}+ \int_0^\infty \frac{\exp\left[-t\sqrt{\xi_1^2+\xi_2^2}\right]}{2\sqrt{\xi_1^2+\xi_2^2}}\, \psi(t,\xi_1,\xi_2)\,dt \tag{9} \]
(we assume that \(\operatorname{Re}\sqrt{\xi_1^2+\xi_2^2}>0\)).
Let us show that, when the conditions of the theorem are fulfilled, \(f(\xi_1,\xi_2)\equiv0\). We shall do this with the aid of Lemmas 1 and 2, for which it will be necessary to estimate the function \(f(z\cos\varphi,z\sin\varphi)\) in the half-plane \(\operatorname{Re}z\geqslant \sigma>0\).
First we obtain an estimate for \(\tilde u(x,z\cos\varphi,z\sin\varphi)\). From condition (3) it follows that this function is an entire function of the variable \(z\) and that it satisfies the inequality
\[ \left|u'(x,(s\pm it)\cos\varphi,(s\pm it)\sin\varphi)\right| \leqslant M\exp\left[\exp\left(\frac{\pi}{h+\eta}x\right)+t h_{\mp}(\varphi)\right], \quad \tau\geqslant0, \tag{10} \]
where
\[ h_+(\varphi)=\max_{(x_1,x_2)\subset D}\{x_1\cos\varphi+x_2\sin\varphi\};\qquad h_-(\varphi)=\min_{(x_1,x_2)\subset D}\{x_1\cos\varphi+x_2\sin\varphi\}; \]
similarly, from condition (2) we obtain the inequality
\[ \left|\psi(x,(s\pm it)\cos\varphi,(s\pm it)\sin\varphi)\right| \leqslant M\exp\left[t h_{\mp}(\varphi)\right], \quad \tau\geqslant0. \tag{11} \]
Inequalities (10) and (11) make it possible to obtain from formulas (8) and (9) estimates for the functions \(\Phi\) and \(f\):
\[ |\Phi(x,s\cos\varphi,s\sin\varphi)|\leqslant M,\qquad s\geqslant\sigma,\ x>0; \tag{12} \]
\[ |f((s\pm it)\cos\varphi,(s\pm it)\sin\varphi)| \leqslant M\exp\left[t h_{\mp}(\varphi)\right], \quad \tau\geqslant0,\ s\geqslant\sigma. \tag{13} \]
Now from formula (7) and inequalities (10) and (12) we find for \(f(z\cos\varphi,z\sin\varphi)\) the stronger estimate, for real \(z=s\geqslant\sigma>0\),
\[ |f(s\cos\varphi,s\sin\varphi)|\leqslant \tag{14} \]
\[ \leqslant \min_{x>0}\left| \tilde u(x,s\cos\varphi,s\sin\varphi) -\Phi(x,s\cos\varphi,s\sin\varphi) \right|\exp[-xs]\leqslant \]
\[ \leqslant \min_{x>0} M\exp\left(\exp\left[\frac{\pi}{h+\eta}x\right]-sx\right) < M_1\exp\left[-\frac{h+\eta_1}{\pi}s\ln s\right], \quad \eta_1>0,\ s\geqslant\sigma. \]
It remains to note that \(\min_\varphi\{h_+(\varphi)-h_-(\varphi)\}=h\); therefore there exists an interval \([\varphi_0-\delta,\varphi_0+\delta]\) of values of \(\varphi\) for which the inequality
\[ h+\eta_1>h_+(\varphi)-h_-(\varphi) \]
holds.
According to estimates (13) and (14), it follows from this that, for any fixed \(\varphi\) from the indicated interval, the functions \(f_\varphi(z)=f(z\cos\varphi,z\sin\varphi)\) satisfy all the conditions of Lemma 1. Hence the function \(f(\xi_1,\xi_2)\) satisfies the conditions of Lemma 2, i.e. \(f(\xi_1,\xi_2)\equiv0\).
On the basis of this relation, from formula (7) we obtain
\[ \tilde u(x,\xi_1,\xi_2)= \left\{ \frac12\tilde u(0,\xi_1,\xi_2)- \frac{\tilde u'_x(0,\xi_1,\xi_2)}{2\sqrt{\xi_1^2+\xi_2^2}} \right\} \exp\left[-x\sqrt{\xi_1^2+\xi_2^2}\right]- \]
\[ -\int_0^\infty \frac{\exp\left[-\sqrt{\xi_1^2+\xi_2^2}\,|x-t|\right]}{2\sqrt{\xi_1^2+\xi_2^2}}\, \psi(t,\xi_1,\xi_2)\,dt. \]
Substituting the expressions for $\widetilde u(0,\xi_1,\xi_2)$, $\widetilde u_x'(0,\xi_1,\xi_2)$ and $\psi(t,\xi_1,\xi_2)$ into (7) and applying the inverse Fourier transform, we find
\[ u(x,x_1,x_2)= \]
\[ =\frac{1}{4\pi^2} \int_{-\infty}^{\infty}\iint_D \left\{ \frac12 u(0,t_1,t_2) - \frac{u_x'(0,t_1,t_2)}{2\sqrt{\xi_1^2+\xi_2^2}} \right\} \exp\left[i\xi_1(t_1-x_1)+i\xi_2(t_2-x_2)-\right. \]
\[ \left. -\,x\sqrt{\xi_1^2+\xi_2^2} \right]\,dt_1dt_2d\xi_1d\xi_2+ \]
\[ +\frac{1}{4\pi^2} \int_{-\infty}^{\infty}\int_0^\infty\int_\gamma u(t,t_1,t_2)\exp\left[i\xi_1(t_1-x_1)+i\xi_2(t_2-x_2)-\right. \]
\[ \left. -|x-t|\sqrt{\xi_1^2+\xi_2^2} \right] \frac{\xi_2dt_1-\xi_1dt_2}{2\sqrt{\xi_1^2+\xi_2^2}}\,dtd\xi_1d\xi_2- \]
\[ -\frac{1}{4\pi^2} \int_{-\infty}^{\infty}\int_0^\infty\int_\gamma \frac{1}{2\sqrt{\xi_1^2+\xi_2^2}} \exp\left[i\xi_1(t_1-x_1)+i\xi_2(t_2-x_2)-\right. \]
\[ \left. -|x-t|\sqrt{\xi_1^2+\xi_2^2} \right] \left(\frac{\partial u}{\partial t_2}\,dt_1-\frac{\partial u}{\partial t_1}\,dt_2\right)dtd\xi_1d\xi_2. \]
Changing the order of integration (the legitimacy of this change is easily verified directly by a limiting passage) and observing that
\[ \int_{-\infty}^{\infty} \frac{1}{2\sqrt{\xi_1^2+\xi_2^2}} \exp\left[i\xi_1(t_1-x_1)+i\xi_2(t_2-x_2)-|x-t|\sqrt{\xi_1^2+\xi_2^2}\right]d\xi_1d\xi_2= \]
\[ = \frac{\pi}{\sqrt{(t-x)^2+(t_1-x_1)^2+(t_2-x_2)^2}} = \frac{\pi}{r}, \]
we arrive at formula (4). The theorem is proved.
The sharpness of the constant $\pi/h$ found for the case when the domain is a disk or a rectangle was proved in $(^{1,2})$. Apparently, the constant $\pi/h$ is sharp also for a broader class of domains; however, the proof of this fact requires a special construction.
The method used above can be successfully applied not only to the Laplace equation, but also to arbitrary elliptic systems with constant coefficients. We have preferred to restrict ourselves only to harmonic functions in order to be able to refer to the well-known estimates for single- and double-layer potentials.
Received
26 VIII 1961
REFERENCES
- M. A. Evgrafov, I. A. Chegis, DAN, 134, No. 2 (1960).
- I. A. Chegis, DAN, 136, No. 3 (1960).
- V. I. Smirnov, Course of Higher Mathematics, 4, 2nd ed., 1951.