MATHEMATICS

E. M. LANDIS

A THEOREM ON THREE BALLS

(Presented by Academician I. G. Petrovskii on July 7, 1962)

In this note Hadamard’s theorem on three circles for analytic functions of a complex variable \(\left({}^{1}\right.\), p. 469) is generalized to solutions of linear elliptic equations of the second order with many independent variables.

We consider the equation

\[ \sum_{i,k=1}^{n} a_{ik}(x)\frac{\partial^{2}u}{\partial x_i\partial x_k} +\sum_{i=1}^{n} b_i(x)\frac{\partial u}{\partial x_i} +c(x)u=0, \tag{1} \]

whose coefficients, throughout the domain where the equation is defined, satisfy the conditions

\[ \sum_{i,k=1}^{n} a_{ik}(x)\xi_i\xi_k \geq \alpha \sum_{i=1}^{n}\xi_i^{2},\qquad \alpha>0; \tag{2} \]

all coefficients are bounded in absolute value by a constant \(M\); the coefficients \(a_{ik}\) are twice continuously differentiable and all their first and second partial derivatives are bounded in absolute value by the constant \(M\); the remaining coefficients are continuously differentiable and their derivatives are bounded by the same constant \(M\); and, finally,

\[ c(x)\leq 0. \tag{3} \]

Theorem. Let in the ball \(Q\) of radius \(r_2<1\) with center at the origin there be defined a solution \(u(x)\) of equation (1), continuous in the closed ball. Denote

\[ M(r)=\max_{|x|=r}|u(x)|. \]

Then for any \(r_1\) and \(r\), \(0<r_1<r<r_2\), the inequality

\[ \ln M(r)\leq \ln M(r_1)\frac{\ln Cr/r_2}{\ln r_1/r_2} +\ln M(r_2)\frac{\ln Cr/r_1}{\ln r_2/r_1} +\ln\ln\frac{C}{r}, \tag{4} \]

holds, where \(C\) is a constant depending on the constant \(\alpha\) in inequality (2), on \(M\), and on the dimension \(n\) of the space.

The proof is close in idea to the proofs of the uniqueness theorem for the solution of the Cauchy problem for an elliptic equation given by Heinz \(\left({}^{2}\right)\) and Cordes \(\left({}^{3}\right)\).

The presence in the right-hand side of inequality (4) of the additional term \(\ln\ln \dfrac{C}{r}\) is possibly connected with the method of proof. Condition (3) is inessential and has been introduced to simplify the proof.

We outline the main steps of the proof.

\(1^\circ\). It is enough to prove the following assertion:

Let in the ball \(Q_1\) of radius 1 there be defined a solution \(u(x)\) of equation (1), continuous in the closed ball. Let \(M(1)\leq 1\), and let, for some \(r_1\), \(0<r_1<1\), \(M(r_1)=r_1^{\beta}\), \(\beta>0\).

Then for every \(r\), \(r_1<r<1\), the inequality

\[ M(r)\leq (Cr)^\beta \ln \frac{C}{r}, \tag{5} \]

holds, where \(C\) is a constant depending on \(\alpha\), \(M\), and \(n\).

\(2^\circ\). From Cordes’ results \((^3)\) it follows that it is enough to prove the assertion of item \(1^\circ\) for the equation

\[ Lu\equiv \frac{\partial^2 u}{\partial r^2}+\frac{n-1}{r}\frac{\partial u}{\partial r} +\frac{1}{r^2}M_r u+\sum_{i=1}^n b_i(x)\frac{\partial u}{\partial x_i} +c(x)u=0, \]

where \(M_r\), for each fixed \(r\), is a linear elliptic self-adjoint operator on the unit sphere \(K_1\), satisfying the following condition: for every function \(\omega\), defined and twice continuously differentiable on the sphere \(K_1\), the inequality
\[ \frac{d}{dr}\int_{K_1}\omega M_r\omega\,d\sigma_1\leq 0 \]
holds, where \(d\sigma_1\) is the surface element of the unit sphere \(K_1\).

\(3^\circ\). Let \(f(r)\) be some fixed twice continuously differentiable function on \([1/2,\,3/4]\), having the properties:

\[ 0\leq f(r)\leq 1,\qquad f(1/2)=1,\qquad f(3/4)=0, \]
\[ f'(1/2)=f''(1/2)=f'(3/4)=f''(3/4)=0. \]

Fix some \(r_0\), \(r_1<r_0<1\), and put

\[ v(x)= \begin{cases} u(x), & \text{for } \frac12 r_1\leq |x|\leq \frac12 r_0,\\ u(x)f\!\left(\frac{1}{r_0}|x|\right), & \text{for } \frac12 r_0< |x|\leq \frac34 r_0,\\ 0, & \text{for } \frac34 r_0< |x|\leq r_0. \end{cases} \]

Then for the function \(v(x)\), by the maximum principle and Bernstein’s inequality \((^4)\), we have

\[ |v|_{|x|=r_1/2}\leq r_1^\beta;\qquad \sum_{i=1}^n\left|\frac{\partial v}{\partial x_i}\right|_{|x|=r_1/2} < C_1(2r_1)^{\beta-1}; \]

\[ \sum_{i,k=1}^n \left|\frac{\partial^2 v}{\partial x_i\partial x_k}\right|_{|x|=r_1/2} < C_1(2r_1)^{\beta-2}; \tag{6} \]

\[ Lv(x)=0\qquad \text{for } \frac12 r_1\leq |x|\leq \frac12 r_0, \tag{7} \]

\[ |Lv(x)|<\frac{C_2}{r_0^2}\qquad \text{for } \frac12 r_0< |x|<r_0. \]

The constants \(C\), provided with indices, here and below will denote constants depending only on \(\alpha\), \(M\), and \(n\).

\(4^\circ\). Put

\[ \frac{\partial}{\partial r^2}+\frac{n-1}{r}\frac{\partial}{\partial r} +\frac{1}{r^2}M_r\equiv L_1. \]

It can be shown, using (6), that

\[ \int_{\frac12 r_1<|x|<r_0} \frac{v^2}{r^{2\beta+n}\ln^2\frac{1}{r}}\,dx \leq \frac{C_3}{\beta^2} \int_{\frac12 r_1<|x|<r_0} \frac{(L_1v)^2}{r^{2\beta+n-4}}\,dx +C_4\beta. \tag{8} \]

Further it is shown that

\[ \int_{\frac12 r_1<|x|<r_0}\frac{|\operatorname{grad} v|^2}{r^{2\beta+n-4}}\,dx \le C_5 r_0 \int_{\frac12 r_1<|x|<r_0}\frac{(L_1v)^2}{r^{2\beta+n-4}}\,dx + C_6\beta^3 . \tag{9} \]

We can choose \(r_0\) so that from (8) and (9) we obtain

\[ \int_{\frac12 r_1<|x|<r_0} \frac{v^2}{r^{2\beta+n}\ln^2\frac1r}\,dx \le \frac{C_7}{\beta^2} \int_{\frac12 r_1<|x|<r_0} \frac{(Lv)^2}{r^{2\beta+n-4}}\,dx + C_8\beta . \]

Here the choice of \(r_0\) depends on the constants \(C_i\), i.e., on \(\alpha\), \(M\), and \(n\). Applying inequality (7), we obtain

\[ \int_{\frac12 r_1<|x|<r_0} \frac{v^2}{r^{2\beta+n}\ln^2\frac1r}\,dx \le \frac{C_9}{\beta^2 r_0^{2\beta-2}} + C_{10}\beta . \tag{10} \]

\(5^\circ.\) Suppose that \(\beta>1\), \(r_0<\frac12\), and let \(r\), \(\frac12 r_1<2r<r_0\), be an arbitrary number. Then from (10)

\[ \frac{1}{(2r)^n} \int_{\frac12 r_1<|x|<2r} v^2\,dx \le C_{11}\beta\left(\frac{2r}{r_0}\right)^{2\beta}\ln^2\frac1r . \]

But from this inequality it follows (see (5)) that

\[ |v|_{|x|=r} \le C_{12}\left(\frac{2r}{r_0}\right)^\beta \ln\frac1r \le (C_{13}r)^\beta \ln\frac1r \]

or

\[ |u|_{|x|=r}\le (C_{13}r)^\beta \ln\frac1r \tag{11} \]

for all \(r\), \(r_1<r<\frac12 r_0\), and \(\beta>1\).

Since for \(\beta\le1\) the inequality (5) that we need is obtained at once from the fact that \(|\operatorname{grad} u|_{|x|<1/2}<C_{15}\), while for \(r\ge \frac12 r_0\) and sufficiently large \(C_{14}\) inequality (5) is obvious, it follows that for every \(r\), \(r_1<r<1\), we have

\[ M(r)<(C_{14}r)^\beta \ln\frac1r, \]

as was required to prove.

Moscow State University
named after M. V. Lomonosov

Received
5 VII 1962

REFERENCES

\({}^{1}\) A. I. Markushevich, Theory of Analytic Functions, M.—L., 1950.
\({}^{2}\) E. Heinz, Nachr. Akad. Wiss. Göttingen, No. 1 (1955).
\({}^{3}\) H. Cordes, Nachr. Akad. Wiss. Göttingen, No. 11 (1956).
\({}^{4}\) S. N. Bernstein, Collected Works, 3, Publishing House of the Academy of Sciences of the USSR, 1960.
\({}^{5}\) K. Miranda, Equations with Partial Derivatives of Elliptic Type, IL, 1957.