UDC 517.947.42

MATHEMATICS

M. L. GERVER

METRIC PROPERTIES OF HARMONIC FUNCTIONS

(Presented by Academician I. G. Petrovskii, September 30, 1965)

In this note we shall formulate two theorems on harmonic functions of two variables. An imprecise, but brief, formulation of Theorem 1 is: harmonic functions have few long level lines.

The exact formulation and the outline of the proof are given in § 1. In Theorem 2 a bounded harmonic function that is small on a set of sufficiently large measure is estimated. This theorem is a generalization of Hadamard’s theorem on three circles. Both theorems admit a generalization to the case of elliptic equations with variable coefficients. Theorem 1 is readily carried over to the multidimensional case. Theorem 2 (both in the plane and in space) can also be obtained by a completely different method as a consequence of the results of M. M. Lavrent’ev \((^4)\).

Throughout what follows, \(K_R\) denotes the disk \(x^2 + y^2 < R^2\).

§ 1. On the length of a level line. Let \(u(x,y)\) be a bounded function harmonic in \(K_R\). The set of points \((x,y) \in K_{R/2}\) at which \(u(x,y)=t\) will be denoted by \(E_t\). For any \(t\), the set \(E_t\) is the union of a finite or countable number of pairwise nonintersecting analytic simple arcs. We denote the sum of the lengths of these arcs by \(l(t)\). Let \(n\) be a natural number. The set of those \(t\) for which \(l(t)>16Rn^2\) will be denoted by \(B_{u,n}\). Let \(U_M\) be the set of all functions \(u\) harmonic in \(K_R\) and bounded in absolute value by the constant \(M\).

Theorem 1. For every \(M>0\) there exists an \(N\) such that, for any \(n>N\) and any function \(u \in U_M\), the measure of \(B_{u,n}\) is less than \(2^{-n}\).

The example of the function \(r^n \sin n\varphi\) in the disk \(K_1\) shows that this estimate cannot be greatly improved. The method of proof of Theorem 1 is close to that used in \((^1)\).

We outline the proof. Let \(u \in U_M\). Fix \(t \in B_{u,n}\) so that \(E_t\) contains no zeros of \(\operatorname{grad} u\). Let \((x,y) \in E_t\). If the tangent to \(E_t\) at the point \((x,y)\) forms with the \(x\)-axis an angle not exceeding \(\pi/4\), assign the point \((x,y)\) to the set \(X\). The set \(E_t \setminus X\) will be denoted by \(Y\).

From \(X\) we select a subset \(X_1\) by means of the following condition. A point \((x,y) \in X\) is assigned to \(X_1\) if there exists a segment of length \(R/2n\), parallel to the \(y\)-axis, containing the point \((x,y)\) and intersecting \(E_t\) in at least \(n\) points. Similarly, if some segment of length \(R/2n\), parallel to the \(x\)-axis and passing through a point \((x,y) \in Y\), intersects \(E_t\) in at least \(n\) points, then we assign \((x,y)\) to the set \(Y_1\).

The proof of Theorem 1 is carried out in three stages:

1. It is established that at the points of the set \(X_1 \cup Y_1\), for \(n>N(M)\), the inequality
   \[ |\operatorname{grad} u| < \frac{2^{-n}}{R} \]
   holds.

2. It is verified that the linear measure of \(X_1 \cup Y_1\) is greater than \(l(t)/2\).

3. From 1 and 2 it follows that
   \[ \operatorname{mes} B_{u,n} < 2^{-n}. \]

§ 2. A generalization of the three-circles theorem. One of the possible formulations of the theorem on three circles for harmonic functions is as follows*:

* This theorem can easily be obtained from the well-known Hadamard theorem on three circles for analytic functions of a complex variable \((^2)\).

If \(u(x,y)\) is a function harmonic in \(K_1\), and if \(|u|<1\) in \(K_1\), \(|u|<\varepsilon^2\) in \(K_{r^2}\), then \(|u|<C\varepsilon\) in \(K_r\) (\(\varepsilon\) and \(r\) are arbitrary numbers between \(0\) and \(1/2\), \(C\) is an absolute constant).

How does this estimate change if, in the formulation of the theorem, the circle \(K_{r^2}\) is replaced by an arbitrary set \(\mathfrak M\) of the same measure \((\operatorname{mes}_2 \mathfrak M=\pi r^4)\)?

The answer to this question is given by

Theorem 2. There exists an \(s>0\) such that for every set \(\mathfrak M\) of measure \(\pi r^4\), situated in \(K_r\), from the inequalities
\[ |u|<1 \text{ in } K_1,\qquad |u|<\varepsilon^s \text{ on } \mathfrak M \]
it follows that
\[ |u|<\varepsilon \text{ in } K_r. \]
(As above, \(\Delta u=0\) in \(K_1,\ 0<\varepsilon,r<1/2\).)

Let us note that it is enough to prove Theorem 2 for \(r=r_0\), in order to be sure of its validity for \(r=r_0^n\) (where \(n\) is any natural number). The case \(r=r_0^n\) is reduced to the case \(r=r_0\) if one maps the circle \(K_1\) onto itself by means of the transformation \(\sqrt[n]{z}\) \((z=x+iy)\).

Consequently, it is enough to establish that the following is true.

Theorem \(2'\). There exists an \(s>0\) such that for every set \(\mathfrak M\) of measure \(\pi/16\), situated in \(K_{1/2}\), for every function \(u(x,y)\) harmonic in \(K_1\), and for every sufficiently large natural number \(N\), from the inequalities
\[ |u|<1 \text{ in } K_1,\qquad |u|<2^{-sN} \text{ on } \mathfrak M \]
it follows that
\[ |u|<2^{-N} \text{ in } K_{1/2}. \]

We outline the plan of the proof of Theorem \(2'\).

1. We shall call the following construction the \(k\)-division of the circle \(K_R\). Consider \(k\) concentric circumferences with center \(0\) and radii \(R/k, 2R/k,\ldots,iR/k,\ldots,R\). Denote by \(H_2,\ldots,H_k\) the concentric rings of width \(R/k\) enclosed between these circumferences. We divide the ring \(H_i\) \((1<i\le k)\) into \(2i-1\) equal regions \(D_{i1},\ldots,D_{i,2i-1}\); to \(D_{ij}\) belong those points \((r,\varphi)\) of the ring \(H_i\) for which
   \[ \frac{2\pi}{2i-1}(j-1)<\varphi<\frac{2\pi}{2i-1}j,\qquad 1\le j\le 2i-1. \]
   All the regions \(D_{ij}\), obviously, are of equal area: \(\operatorname{mes}_2 D_{ij}=\pi R^2/k^2\). We shall call \(H_2,\ldots,H_k\) the rings of the \(k\)-division, and \(D_{21},\ldots,D_{k,2k-1}\) the cells of the \(k\)-division.

2. By the hypothesis of the theorem, \(|u|<2^{-sN}\) on \(\mathfrak M\). Denote by \(\mathfrak M_1\) the set of points of \(K_{1/2}\) where \(|u|<2^{1-sN}\). With the help of Theorem 1 it is not difficult to prove that there exists an absolute constant \(k_0\) with the following property:

Let \(k\ge k_0(sN)^2\). Carry out the \(k\)-division of the circle \(K_{1/2}\). Then at least \(k^2/8\) cells of the division belong entirely to \(\mathfrak M_1\).

1. Now, using the theorem on three circles, one can reduce Theorem \(2'\) to the following lemma:

Lemma 1. Let \(q\) and \(n\) be natural numbers. Carry out the \(2^{q+4}\)-division of the circle \(K_{1/2}\). Let \(u(x,y)\) be a harmonic function in the circle \(K_1\), satisfying the conditions: 1) \(|u|<1\) in \(K_1\); 2) among the rings of the division situated between the circumferences of radii \(1/32\) and \(1/2\), there are \(2q\) or more rings containing cells of the division where \(|u|<2^{-2q+n}\) (at least one such cell in each); 3) \(\max_{K_{1/2}} |u|>2^{-2q}\).

Then \(n<A\), where \(A\) is an absolute constant.

The proof of Lemma 1 recalls the proof of Lemma 1.6.1 in (3).

Moscow State University
named after M. V. Lomonosov

Received
29 IX 1965

REFERENCES

1. E. M. Landis, DAN, 79, No. 3, 393 (1951).
2. A. I. Markushevich, Theory of analytic functions, M.—L., 1950.
3. E. M. Landis, UMN, 14, issue 1 (85), 21 (1959).
4. M. M. Lavrent’ev, On certain incorrect problems of mathematical physics, Novosibirsk, 1962.