M. L. Gerver, E. M. Landis

A Generalization of the Mean Value Theorem for Functions of Several Variables

(Presented by Academician I. G. Petrovskii on 17 IV 1962)

The proposed theorem was needed in solving a number of questions in the theory of partial differential equations. It may be regarded as a certain analogue of the mean value theorem for functions of \(n\) variables, \(n \ge 2\).

Let \(G\) be a bounded domain of the \(n\)-dimensional space of variables \(x_1,\ldots,x_n\). The measure of the closed domain \(\overline{G}\) is less than 1: \(\operatorname{mes}\overline{G}<1\). The domain \(G\) is situated in the strip
\[ \Pi\{a<x_1<b;\ -\infty<x_k<\infty,\ k\ge 2\}, \]
\(a<0;\ b>1\). The intersection of the boundary \(\Gamma\) of the domain \(G\) with the \((n-1)\)-dimensional planes \(x_1=a\) and \(x_1=b\) is nonempty in both cases. Denote them respectively by \(\Gamma_a\) and \(\Gamma_b\).

We shall say that a set \(M\) separates \(\Gamma_a\) from \(\Gamma_b\) in \(G\) if any polygonal line \(L \subset G\cup \Gamma_a\cup \Gamma_b\), \(L\cap \Gamma_a\ne 0\), \(L\cap \Gamma_b\ne 0\), intersects \(M\): \(L\cap M\ne 0\). In the closed domain \(\overline{G}\) a twice continuously differentiable function \(f(x)=f(x_1,\ldots,x_n)\) is given, \(\operatorname{osc} f(x)<1\).

It is required to prove that

there exists a finite number of smooth surfaces \(S_1,\ldots,S_k\) such that
\[ \bigcup_{i=1}^{k} S_i \text{ separates } \Gamma_a \text{ from } \Gamma_b \text{ in } G \quad \text{and} \quad \sum_{i=1}^{k}\int_{S_i\cap G}\left|\frac{\partial f}{\partial n}\right|\,ds<8 \]
(\(n\) is the normal to the surface).

Proof. First of all note that \(\left|\partial f/\partial n\right|\) is the length of the projection onto the direction \(n\) of the vector \(\operatorname{grad} f\). Therefore, if at every point of an \((n-1)\)-dimensional surface \(S\) the vector \(n\)—the normal to the surface \(S\)—and the vector \(\operatorname{grad} f\) are orthogonal, then
\[ \int_S \left|\frac{\partial f}{\partial n}\right|\,ds=0. \]

In what follows it will be convenient for us to assume that the function \(f(x)\) is given not only in \(\overline{G}\), but in a wider domain \(\overline{G_\alpha}\). More precisely, let \(G_\alpha\) be the \(\alpha\)-expansion of the domain \(G\), and let \(\alpha\) be so small that \(\operatorname{mes}\overline{G_\alpha}<1\); extend the function \(f(x)\) to \(G_\alpha\) in a twice continuously differentiable way, preserving the inequality \(\operatorname{osc} f<1\), and so that in some neighborhood of the boundary \(\Gamma_a\) of the domain \(G_\alpha\) one has \(\operatorname{grad} f\ne 0\).

Consider in \(\overline{G_\alpha}\) the field \(\operatorname{grad} f\). We shall prove (in a special lemma) that the set of its singular points can be covered by a finite number of balls \(\widetilde{\Omega}_i\), \(i=1,\ldots,l\), such that for some \(\beta\), \(\alpha>\beta>0\), the \(\beta\)-expansions of these balls \(\Omega_i\), \(i=1,\ldots,l\), belong to \(G_\alpha\) and
\[ \sum_{i=1}^{l}\int_{\omega_i}|\operatorname{grad} f|\,ds<1, \]
where
\[ \omega_i=\overline{\widetilde{\Omega}_i}\setminus \Omega_i. \]

Introduce the following notation:
\[ \widetilde{D}=G_\alpha\setminus \bigcup_{i=1}^{k}\widetilde{\Omega}_i; \]
\(\widetilde{\Delta}\) is the boundary of \(\widetilde{D}\); \(\widetilde{l}_Q\) is the trajectory of the field \(\operatorname{grad} f\) in \(\widetilde{D}\) passing through the point \(Q\in\widetilde{D}\); \(D\) is the set of points \(x\in\widetilde{D}\) whose distance from \(\widetilde{\Delta}\) is greater than \(\beta\); \(\Delta\) is the boundary of \(D\); \(l_Q\) is the trajectory of the field \(\operatorname{grad} f\) in \(D\) passing through the point \(Q\in D\).

We now set forth the plan of the proof. We shall call a domain \(P \subset D\) a bundle of trajectories if \(P\), together with each of its points \(Q\), contains the entire trajectory \(l_Q\). The part \(T\) of the boundary of the domain \(P\) consisting of points \(Q \notin \Delta\) obviously has the same property. We shall call \(T\) a tube of trajectories. Obviously,

\[ \int_T \left| \frac{\partial f}{\partial n} \right|\, ds = 0. \]

We shall call a tube \(T\) through if there exists a broken line \(L \subset P\) connecting \(\Gamma_a\) and \(\Gamma_b\). Suppose that we have succeeded in finding a finite number of bundles of trajectories \(P_1,\ldots,P_m\) satisfying the requirements:

1) \(\displaystyle \bigcup_i \overline{P_i} \supset D\); 2) each through tube \(T_i\) can be “partitioned” by an \((n-1)\)-dimensional plane in such a way that the part of the plane that falls inside \(T_i\) (the “partition” \(\pi_i\)) separates \(\Gamma_a\) from \(\Gamma_b\) inside \(T_i\), and moreover

\[ \sum_i \int_{\pi_i} |\operatorname{grad} f|\, ds < 6. \]

Then the union of all tubes (including non-through ones), all partitions \(\pi_i\), and all spheres \(\omega_j\), \(j=1,\ldots,l\), is the desired set \(M\).

Bundles \(P_i\) satisfying requirements 1), 2) can indeed be found. But the proof of this fact is rather long. We shall construct not a finite, but a countable set of bundles \(P_i\), \(\overline{P_i}\cap P_j=0\) for \(i\ne j\), satisfying requirements 1), 2). For any \(\varepsilon>0\) one can choose from our countable set of bundles a finite subset

\[ P_1,\ldots,P_m, \]

such that the points

\[ x \in D \setminus \bigcup_{i=1}^{m} \overline{P_i}, \]

for which \(x_1=1/2\), form a set \(S_0\) whose \((n-1)\)-dimensional measure is less than \(\varepsilon\): \(\operatorname{mes}_{n-1} S_0<\varepsilon\). Let \(\tau_i\) be the intersection of \(P_i\) with the plane \(x_1=1/2\), \(i=1,\ldots,m\). Denote by \(\tau_i^\gamma\) the set of points \(x\in \tau_i\) whose distance to the boundary of \(\tau_i\) is greater than \(\gamma>0\), and let \(\gamma\) be so small that

\[ \operatorname{mes}_{n-1}(\tau_i\setminus \tau_i^\gamma)<\varepsilon/m. \]

If \(\tau_i^\gamma\) consists of infinitely many components, then we choose from it a finite number of components whose union \(\sigma_i\) differs from \(\tau_i^\gamma\) in measure by less than \(\varepsilon/m\):

\[ \operatorname{mes}_{n-1}(\tau_i^\gamma\setminus \sigma_i)<\varepsilon/m. \]

If \(\tau_i^\gamma\) consists of a finite number of components, put \(\sigma_i=\tau_i^\gamma\). Denote by \(S\) the set of points of the plane \(x_1=1/2\) that do not belong to

\[ \bigcup_{i=1}^{m}\sigma_i \cup \bigcup_{j=1}^{l}\omega_j. \]

For

\[ \varepsilon=\frac{1}{3\max_{x\in \overline{D}}|\operatorname{grad} f(x)|}, \]

it is obvious that

\[ \int_{S\cap G} |\operatorname{grad} f|\, ds < 1. \]

Therefore the sum

\[ S \cup \bigcup_{i=1}^{m} T_i \cup \bigcup_{i=1}^{m} \pi_i \cup \bigcup_{j=1}^{l}\omega_j \]

may be taken as \(M\).

Thus, in order to prove the theorem, it suffices to prove the lemma on the exceptional points of the field \(\operatorname{grad} f\) and to construct a countable set of bundles \(P_i\) satisfying requirements 1), 2).

We begin with the lemma. Let \(N\) be the set of points \(x\in \overline{G_a}\) at which \(\operatorname{grad} f(x)=0\). Represent \(N\) as the sum \(N=N_1\cup N_2\), where \(N_1\) consists of the points \(x\in N\) at which \(d^2 f(x)\ne 0\), and \(N_2\) of the points \(x\in N\) at which \(d^2 f=0\).

I. It is easy to show that all density points of the set \(N_1\) belong to \(N_2\), i.e. \(\operatorname{mes} N_1=0\). Therefore, for any \(\varepsilon>0\) there is an open set \(O_1\) such that \(\operatorname{mes} O_1<\varepsilon\), \(N_1\subset O_1\subset \overline{G_a}\). For each point \(x\in N_1\) construct a ball \(\Omega\) of radius \(r(\Omega)\) so small that \(\Omega\subset O_1\). Denote the set of all such balls by \(M_1\). Let

\[ R_1=\sup_{\Omega\in M_1} r(\Omega). \]

Select from \(M_1\) a finite or countable set of balls according to the following rule. As \(\Omega_1\) we take an arbitrary ball \(\Omega\in M_1\) such that \(r(\Omega)>R_1/2\). Suppose that \(\Omega_1,\ldots,\Omega_{i-1}\) have already been chosen and are pairwise disjoint.

Denote by \(M_i\) the set of balls \(\Omega \in M_1\) that do not intersect any of the balls \(\Omega_1,\ldots,\Omega_{i-1}\). If \(M_i\) is empty, the selection of balls is completed. Otherwise we take for \(\Omega_i\) an arbitrary ball \(\Omega \in M_i\) such that \(r(\Omega)>R_i/2\), where \(R_i=\sup_{\Omega\in M_i} r(\Omega)\). The balls \(\Omega_i\) are pairwise disjoint, so that \(e_n\sum_i r^n(\Omega_i)<\varepsilon\), where \(e_n\) is the volume of the unit \(n\)-dimensional ball.

Next, for each \(i\) we construct balls \(\Omega_i'\) and \(\widetilde{\Omega}_i'\), concentric with \(\Omega_i\), with radii \(r_i'=10r(\Omega_i)\) and \(\widetilde r_i'=5r(\Omega_i)\). It is clear that the balls \(\widetilde{\Omega}_i'\) cover \(N_1\). Moreover, we may assume that the balls \(\Omega\in M\) were chosen so small that \(\Omega_i'\subset G_\alpha\). For every \(i\), at the center of the ball \(\Omega_i'\), \(\operatorname{grad} f=0\). Therefore (in view of the boundedness of the second derivatives \(f(x)\)) on the sphere \(\omega_i=\overline{\Omega}_i'-\Omega_i'\), \(|\operatorname{grad} f|\le cr_i'\), where \(c\) is a constant independent of \(i\).

Thus,
\[ \sum_i \int_{\omega_i'} |\operatorname{grad} f|\,ds \le \sum_i cr_i' \int_{\omega_i'} ds = \sum_i cr_i'\cdot s_n r_i'^{\,n-1}, \]
where \(s_n\) is the area of the surface of the unit \((n-1)\)-dimensional sphere. Consequently,
\[ \sum_i \int_{\omega_i'} |\operatorname{grad} f|\,ds \le cs_n\sum_i r_i'^{\,n} < \frac{10^n cs_n}{e_n}\,\varepsilon < \frac12 \quad\text{for}\quad \varepsilon=\frac{e_n}{2\cdot 10^n cs_n}. \]

II. Let \(x^0=(x_1^0,\ldots,x_n^0)\in N_2\), i.e. \(\operatorname{grad} f(x^0)=d^2f(x^0)=0\). Then for every \(\varepsilon>0\) there is a \(\delta>0\) such that at every point \(x=(x_1,\ldots,x_n)\in G_\alpha\),
\[ \sum_{i=1}^n (x_i-x_i^0)=r^2\le \delta, \]
the inequality \(|\operatorname{grad} f(x)|\le \varepsilon r\) holds, and hence
\[ \int_{\sum_{i=1}^n (x_i-x_i^0)^2=r^2} |\operatorname{grad} f|\,ds \le \varepsilon s_n r^n. \]
Around each point \(x^0\in N_2\) describe a ball \(\Omega\subset G_\alpha\) of radius \(r\) such that
\[ \int_{\sum_{i=1}^n (x_i-x_i^0)^2=r^2} |\operatorname{grad} f|\,ds \le \varepsilon s_n r^n. \]

Let \(\Omega^0\) be the ball concentric with \(\Omega\) of radius \(r^0=r/6\). From the set of balls \(\Omega^0\) choose a finite number of balls \(\Omega_1^0,\ldots,\Omega_s^0\) covering \(N_2\). Further, from this set of balls \(\Omega_1^0,\ldots,\Omega_s^0\), select a subset \(\Omega_1,\ldots,\Omega_j\) according to the following rule: for \(\Omega_1\) take the largest of the balls \(\Omega_i^0\); for \(\Omega_2\), the largest of the balls \(\Omega_i^0\) not intersecting \(\Omega_1\), and so on. The balls \(\Omega_1,\ldots,\Omega_j\) are pairwise disjoint, so that
\[ \sum_{i=1}^j \operatorname{mes}\Omega_i < \operatorname{mes}G_\alpha < 1. \]
For each \(i,\ 1\le i\le j\), consider the balls \(\Omega_i''\) and \(\widetilde{\Omega}_i''\), concentric with \(\Omega_i\), with radii \(r_i''=6r(\Omega_i)\) and \(\widetilde r_i''=3r(\Omega_i)\). It is clear that
\[ \bigcup_{i=1}^j \widetilde{\Omega}_i''\supset N_2,\qquad \sum_{i=1}^j \operatorname{mes}\Omega_i''<6^n, \]
so that, denoting \(\omega_i''=\overline{\Omega}_i''\setminus\Omega_i''\), we find
\[ \sum_{i=1}^j \int_{\omega_i''} |\operatorname{grad} f|\,ds < \varepsilon s_n \sum_{i=1}^j r_i^n < \varepsilon\frac{s_n}{e_n}6^n < \frac12 \quad\text{for}\quad \varepsilon=\frac12\,\frac{e_n}{s_n6^n}. \]

Combining the results of parts I and II, we have constructed a finite or countable set of balls
\[ \widetilde{\Omega}_1',\ldots,\widetilde{\Omega}_i',\ldots,\widetilde{\Omega}_1'',\ldots,\widetilde{\Omega}_j'', \]
covering \(N\), where the balls
\[ \Omega_1',\ldots,\Omega_i',\ldots,\Omega_1'',\ldots,\Omega_j'' \]
(concentric with them and of twice the radii) belong to \(G_\alpha\) and satisfy the condition

\[ \sum_i \int_{\omega_i} |\operatorname{grad} f|\, ds+\sum_{i=1}^l \int_{\omega_i} |\operatorname{grad} f|\, ds<1. \]

The assertion of the lemma follows from this by virtue of the closedness of \(N\).

Let us now pass to the construction of the pencils \(P_i\). Choose \(\delta>0\) so that the following conditions are satisfied:

\(1^\circ.\ \delta<\frac12\min(-a,\, b-1)\).

\(2^\circ.\ \delta<1/\max\limits_{Q\in \widetilde D} k(Q)\), where \(k(Q)\) is the curvature of the trajectory \(\widetilde l_Q\) at the point \(Q\).

\(3^\circ.\) For any two points \(A\) and \(B\in \widetilde D\), the distance between which is less than \(\delta\), the inequality
\[ |\operatorname{grad} f(A)-\operatorname{grad} f(B)|<1 \]
holds.

Let \(\Omega_Q^\varepsilon\) be the open \(n\)-dimensional ball of radius \(\varepsilon\) with center at the point \(Q\), and let \(l_Q^\delta\) be the \(\delta\)-neighborhood of the trajectory \(\widetilde l_Q\). For each point \(A\in \overline D\) construct the ball \(\Omega_A^\varepsilon\). Let \(\varepsilon=\varepsilon(A)>0\) be so small that \(\Omega_A^\varepsilon\subset \widetilde D\), and for any point \(Q\in\Omega_A^\varepsilon\), \(l_Q\subset l_A^\delta\). From the set of balls \(\Omega_A^\varepsilon,\ A\in\overline D\), choose a finite number of balls covering \(\overline D\): \(\Omega^1,\ldots,\Omega^m\).

Let \(C\) be an arbitrary component of the intersection \(\Omega^i\cap D\). The trajectories \(l_Q,\ Q\in C\), occupy a domain \(B\). When the different domains \(B\) intersect, there is formed no more than a countable set of pairwise nonintersecting pencils of trajectories \(P_i\). They obviously satisfy requirement 1). We shall show that requirement 2) is also fulfilled for them.

By construction each pencil \(P_i\) belongs to the \(\delta\)-neighborhood \(l^\delta\) of some trajectory \(\widetilde l=l_i\). If \(T_i\) is a through tube, then by the choice of \(\delta\) (condition \(1^\circ\)) the trajectory \(l_i\) intersects the planes \(x_1=0\) and \(x_1=1\). Let \(l_i'\) be the arc of \(l_i\) enclosed between these planes. Consider
\[ \int_{l_i'} |\operatorname{grad} f|\, dl . \]
Since \(\operatorname{osc} f<1\), it follows that
\[ \int_{l_i'} |\operatorname{grad} f|\, dl<1. \]
Therefore the linear measure of the set \(l_i^2\) of points \(x\in l_i'\) where \(|\operatorname{grad} f(x)|\le 2\) is not less than \(1/2\):
\[ \operatorname{mes}_1 l_i^2\ge \frac12 . \]
Through each point \(x\in l_i^2\) draw the \((n-1)\)-dimensional plane \(\pi_x\) orthogonal to \(l_i\) at the point \(x\). The component of the intersection \(\pi_x\cap P_i\) containing the point \(x\) will be denoted by \(\pi_{xi}\). By the choice of \(\delta\) (condition \(2^\circ\)), \(\pi_{xi}\cap\pi_{yi}=0\) for \(x\ne y\). Each section \(\pi_{xi}\) separates \(\Gamma_a\) from \(\Gamma_b\) inside \(T_i\); as \(\pi_i\) take that one of them whose \((n-1)\)-dimensional measure is minimal:
\[ \operatorname{mes}_{n-1}\pi_i=\min_{x\in l_i^2}\operatorname{mes}_{n-1}\pi_{xi}. \]
We shall prove that
\[ \sum_i \int_{\pi_i} |\operatorname{grad} f|\, ds\le 6 \]
(the summation is over all \(i\) corresponding to through tubes \(T_i\)).

Consider the set
\[ \bigcup_i \bigcup_{x\in l_i^2} \pi_{xi}=\pi . \]
Since \(\pi\subset G_a\), we have \(\operatorname{mes}\pi<1\). On the other hand,
\[ \operatorname{mes}\pi=\sum_i \operatorname{mes}\bigcup_{x\in l_i^2}\pi_{xi} \ge \sum_i \frac12\,\operatorname{mes}_{n-1}\pi_i . \]
Consequently,
\[ \sum_i \operatorname{mes}_{n-1}\pi_i\le 2. \]
Further, by the choice of \(\delta\) (condition \(3^\circ\)), at points \(x\in\pi_i\)
\[ |\operatorname{grad} f(x)|\le 3. \]
Thus,
\[ \sum_i \int_{\pi_i} |\operatorname{grad} f|\, ds\le 6. \]
The theorem is thereby proved.

Moscow State University
named after M. V. Lomonosov

Received
13 IV 1962