UDC 517.947

MATHEMATICS

B. N. KHIMCHENKO

ON A THEOREM OF M. V. KELDYSH AND M. A. LAVRENTIEV

(Presented by Academician A. N. Tikhonov on October 6, 1969)

The central result of paper (¹) is Theorem 2, from which there immediately follows uniqueness (up to a constant) of the solution of the Neumann problem. This theorem is formulated as follows.

Let (u(M)) be a function harmonic in some domain (T) of three-dimensional Euclidean space, not identically constant, and suppose that at a point (M_0) of the boundary (\partial T), (u(M)) has a unique limiting value (u_0), equal to the lower bound of its values in (T).

If in (T) one can inscribe a body congruent to the paraboloid
(z_0 \ge z \ge \rho^{1+\alpha}) ((\alpha > 0;\ \rho = \sqrt{x^2+y^2})) with vertex at (M_0), then

[
\lim_{r_{10}\to 0}\frac{u(M_1)-u_0}{r_{10}}>0,
]

where (M_1) is a point on the axis of the paraboloid, and (r_{10}) is the distance from (M_1) to (M_0).

Let us now introduce the function (\varphi(t)) ((t \in [0,t_0])), satisfying the following conditions:

[
\varphi(t)\in C^{(1)}([0,t_0])\cap C^{(\infty)}((0,t_0]);
\tag{1}
]

[
\varphi(0)=\varphi'(0)=0;
\tag{2}
]

[
\varphi'(t)>0,\ \varphi''(t)>0 \quad \text{in } (0,t_0);
\tag{3}
]

[
\int_0^{t_0}\frac{\varphi(t)\,dt}{t^2}<\infty .
\tag{4}
]

We shall call the body
(z_0 \ge z \ge \varphi(\rho))
(\left(\rho=\left(\sum_{i=1}^{n-1}x_i^2\right)^{1/2}\right))
a (\varphi)-paraboloid.

By constructing a lower barrier, one can prove that Theorem 2 remains valid in a domain ((T+\partial T)) of (n)-dimensional Euclidean space whose boundary point (M_0) can be touched from within by a (\varphi)-paraboloid. As such a barrier one may take the subharmonic function

[
\Psi_1(M)=z\exp\left[\lambda_1\int_0^z\frac{\varphi(t)\,dt}{t^2}\right]-\lambda_2\varphi(r),
]

where
(r=\sqrt{\rho^2+z^2}), and (\lambda_i) (here and below, (\lambda_i) are positive constants).

At the same time, Theorem 2 fails in certain domains ((T+\partial T)\in A^{(1)}).

Indeed, define a harmonic function (u(M)) in a (\varphi)-paraboloid, where (\varphi(t)) satisfies conditions (1)—(3), but

[
\int_0^{t_0}\frac{\varphi(t)\,dt}{t^2}=\infty .
]

Moreover, (u(M)) must attain the minimal value (u_0) in some neighborhood of (M_0). Then on the axis (Oz)

[
u(M_1)-u_0 \leqslant \lambda_3 z \exp\left[-\lambda_4 \int_{z}^{c}\frac{\varphi(t)\,dt}{t^2}\right].
]

As an upper barrier here one proposes the superharmonic function

[
\Psi_2(M)=\exp\left[-\lambda_4 \int_{r}^{r_0}\frac{\varphi(t)\,dt}{t^2}\right]\bigl(\lambda_5 z-\varphi(r)\bigr).
]

I express my sincere gratitude to the participants of A. N. Tikhonov’s seminar for their attention to this work.

Moscow State University
named after M. V. Lomonosov

Received
25 IX 1969

REFERENCES

(^{1}) M. V. Keldysh, M. A. Lavrent’ev, DAN, 16, No. 3 (1937).