MATHEMATICS

V. V. KRIVOV

SOME PROPERTIES OF MODULES IN SPACE

(Presented by Academician M. A. Lavrent’ev on 29 VIII 1963)

Here some properties of modules of families of curves and surfaces in space and of conformal capacity are proved.

1. Let \(G\) be a simply connected domain of \(n\)-dimensional Euclidean space*, on whose boundary \(\partial G\) two nonintersecting continua \(B_0\) and \(B_1\), called the bases of the domain, are marked, such that \(\partial G - (B_0 \cup B_1)\) is connected. The theorems obtained are also valid for a broader class of domains (for example, doubly connected ones with a corresponding choice of bases; see item 6).

By \(\gamma(A,B)\) we shall denote a curve \(\gamma \in G\) joining two sets \(A\) and \(B\); by \(\sigma(A,B)\), a hypersurface \(\sigma \subset G\) separating in \(G\) two sets \(A\) and \(B\).

Let, further, \(\rho \geqslant 0\) be some function (metric) defined in \(G\). We shall assume that \(\rho\) is bounded on every compact set \(F \subset G\), that
\[ \lim_{P \to P_0} \rho < +\infty, \]
if \(P_0 \in \partial G - (B_0 \cup B_1)\), and that \(\rho\) is integrable over any rectifiable curve \(\gamma \subset G\) and over any hypersurface \(\sigma \subset G\) measurable in the sense of Hausdorff measure of order \(n-1\).

Definition. The function
\[ u(P)=\inf_{\gamma(B_0,P)} \int_\gamma \rho\,ds, \]
defined in \(G\), will be called the potential function, or potential, for \(\rho\).

It is easy to show that for any \(\gamma(P_1,P_2)\)
\[ |u(P_1)-u(P_2)| \leqslant \int_{\gamma(P_1,P_2)} \rho\,ds, \]
so that on any compact set \(F \subset G\) \(u(P)\) satisfies the Lipschitz condition. Consequently, almost everywhere \(u(P)\) is differentiable in every direction,
\[ \lim_{P_2\to P_1}\frac{1}{|P_1P_2|}\int_{P_1P_2}\rho\,ds=\rho, \]
where \(P_1P_2\) is the segment joining \(P_1\) and \(P_2\). Thus, from the above inequality we obtain almost everywhere in \(G\)
\[ |\nabla u| \leqslant \rho . \tag{1} \]

1. Denote by \(\{C\}\) the family of all rectifiable curves \(\gamma(B_0,B_1)\), and by \(\{S\}\) the family of all Hausdorff-measurable hypersurfaces \(\sigma(B_0,B_1)\). Put
   \[ h(\rho)=\inf_{\{\gamma\in C\}}\int_\gamma \rho\,ds,\qquad S(\rho)=\inf_{\sigma\in\{S\}}\int_\sigma \rho^{\,n-1}\,dS,\qquad V(\rho)=\int_G \rho^n\,dV. \]
   For example, if \(G\) is a right cylinder, \(B_0\) and \(B_1\) its bases, and \(\rho=1\), then \(h,S,V\) will be, respectively, the height, the area of the base, and the volume of this cylinder.

If \(u\) is the potential of \(\rho\), then
\[ h(\rho)=\inf_{P\in B_1}u(P)\leqslant \sup_{P\in B_1}u(P). \tag{2} \]
It is also easy to show that \(\sigma(a)=\{u=a\}\), for any \(0<a<h(\rho)\), separates \(B_0\) and \(B_1\), and therefore
\[ S(\rho)\leqslant \int_{\sigma(a)} \rho^{\,n-1}\,dS. \tag{2'} \]

* The space is compactified by a point at infinity.

Lemma. If \(\rho_1\) and \(\rho_2\) are two nonnegative metrics, then

\[ \int_G \rho_1^{\,n-1}\rho_2\,dV \geq S(\rho_1)\cdot h(\rho_2); \tag{3} \]

in particular, \(V(\rho)\geq S(\rho)\cdot h(\rho)\).

The proof uses the generalized Fubini theorem from (3), by virtue of which

\[ \int_G \rho_1^{\,n-1}|\nabla u|\,dV = \int_0^m d\alpha \int_{u=\alpha}\rho_1^{\,n-1}dS, \]

where \(m=\sup_{P\in B_1}u(P)\), and as \(u\) one takes the potential for \(\rho_2\). Hence, from (2) and \((2')\), we obtain what is required.

Theorem 1. For the equality

\[ V(\rho)=S(\rho)\cdot h(\rho) \tag{4} \]

to hold, it is necessary and sufficient that the following conditions be satisfied simultaneously*:

1. \(\rho=|\nabla u|\).
2. \(u(P)=\mathrm{const}\), if \(P\in B_1\).
3. \(\displaystyle \int_{\sigma(\alpha)} \rho^{\,n-1}dS=\mathrm{const}\), if \(\sigma(\alpha)=\{u=\alpha\}\).
4. \(\displaystyle \int_{\sigma(\alpha)} \rho^{\,n-1}dS \leq \int_{\sigma}\rho^{\,n-1}dS\) for any \(\sigma\in\{S\}\).

For the proof it is enough to note that, together with (4), the equalities in (1), (2), \((2')\) are satisfied.

1. The modulus \(M\{C\}\) of the family of curves \(\{C\}\) and the modulus \(M\{S\}\) of the family of hypersurfaces \(\{S\}\) can be defined as follows (see \((1,5)\)):

\[ M\{C\}=\inf_{\rho}\frac{V(\rho)}{h^n(\rho)},\qquad M\{S\}=\inf_{\rho}\frac{V(\rho)}{S^{\frac{n}{n-1}}(\rho)}. \tag{5} \]

Theorem 2. If \(V(\rho_0)=S(\rho_0)\cdot h(\rho_0)\), then \(\rho_0\) will be an extremal function both for \(\{C\}\) and for \(\{S\}\), i.e. we shall have

\[ M\{C\}=\frac{V(\rho_0)}{h^n(\rho_0)},\qquad M\{S\}=\frac{V(\rho_0)}{S^{\frac{n}{n-1}}(\rho_0)}. \tag{6} \]

Indeed, the integral \(\displaystyle \int_G \rho\rho_0^{\,n-1}dV\), with the aid of Hölder’s inequality, can be estimated from above:

\[ \int_G \rho\rho_0^{\,n-1}dV \leq V^{\frac1n}(\rho)\cdot V^{\frac{n-1}{n}}(\rho_0), \tag{7} \]

and with the aid of the lemma—from below. After simple transformations we obtain

\[ \frac{V^{\frac1n}(\rho_0)}{h(\rho_0)} \leq \frac{V^{\frac1n}(\rho)}{h(\rho)}, \]

whence the first of the equalities (6) follows. The second is proved analogously with the aid of estimates of the integral \(\displaystyle \int_G \rho_0\rho^{\,n-1}dV\).

* Here and below it is assumed that \(\rho>0\) almost everywhere. Two functions that coincide almost everywhere are not distinguished.

If, together with \(\rho_0\), another function \(\rho\) satisfies equation (4), then in this reasoning everywhere, including in (7), there will be equalities, whence it follows that \(\rho=\lambda \rho_0\), \(\lambda=\mathrm{const}\). Thus, by equation (4), \(\rho\) is determined uniquely up to a numerical factor.

Corollary 1. \(M\{C\}\) and \(M\{S\}\) are related by the relation\(^*\)

\[ M\{C\}M^{n-1}\{S\}=1. \tag{8} \]

This is obtained at once from (6) and (4). From this corollary and from (5) follows

Corollary 2. The estimates hold

\[ \frac{S^n(\rho)}{V^{n-1}(\rho)} \leqslant M\{C\} \leqslant \frac{V(\rho)}{h^n(\rho)}; \tag{9} \]

\[ \frac{h^{\frac{n}{n-1}}(\rho)}{V^{\frac{n}{n-1}}(\rho)} \leqslant M\{S\} \leqslant \frac{V(\rho)}{S^{\frac{n}{n-1}}(\rho)}, \tag{9′} \]

in each of which equality is attained only for the extremal function.

4. The conformal capacity \(\Gamma(G)\) is introduced by the formula

\[ \Gamma(G)=\inf_u \int_G |\nabla u|^n dV, \]

where the greatest lower bound is taken over all continuously differentiable functions such that

\[ u(P)=0 \quad \text{for } P\in B_0; \qquad u(P)=1 \quad \text{for } P\in B_1. \tag{10} \]

If for one of such functions \(\Gamma(G)=\int_G |\nabla u|^n dV\), then \(u\) satisfies the Euler equation for the functional \(\int_G |\nabla u|^n dV\):

\[ \operatorname{div}\bigl(|\nabla u|^{\,n-2}\nabla u\bigr)=0 \tag{11} \]

and the boundary conditions

\[ u(P)=0,\quad P\in B_0; \qquad u(P)=1,\quad P\in B_1; \]

\[ \frac{\partial u}{\partial \nu}=0,\quad P\in \partial G-(B_0\cup B_1) \tag{12} \]

with the corresponding assumptions on the existence of the normal \(\nu\) on the boundary \(\partial G\).

Theorem 3. The modulus of the family of curves \(\{C\}\) is equal to the conformal capacity of the domain \(G\): \(M\{C\}=\Gamma\{G\}\).

Indeed, putting \(\rho=|\nabla u|\), where \(u\) satisfies conditions (10), we obtain \(M\{C\}\leqslant \Gamma(G)\). If, however, \(\rho\geqslant 0\) is some metric and \(u\) is its potential, then, taking \(u_0=u/h(\rho)\), we obtain, in view of (1), \(\Gamma(G)\leqslant M\{C\}\). The theorem is proved.

Theorem 4. If \(\rho_0\) is an extremal metric for \(\{C\}\) (respectively \(\{S\}\)), then it is extremal also for \(\{S\}\) (respectively \(\{C\}\)), and

\[ V(\rho_0)=S(\rho_0)\cdot h(\rho_0). \]

Indeed, let \(u\) be the potential of \(\rho_0\). It then follows from (11) and (12) that

\[ V(|\nabla u|)=S(|\nabla u|)\cdot h(|\nabla u|), \tag{13} \]

and the application of theorem 2 completes the proof.

Thus, the extremal metric both for \(\{C\}\) and for \(\{S\}\) is determined uniquely up to a numerical factor. We shall simply call such a metric extremal (for \(G\)).

Remark. Equation (13) holds if \(u\) satisfies equation (11) and conditions (12) and if every level surface of \(u\)

\(^*\) This relation and theorem 3 were obtained by B. V. Shabat by another method (unpublished).

separates \(B_0\) and \(B_1\). Thus (13) may be regarded as an integral form for (11) and (12).

1. The following two theorems extend to space the known principles of Grötzsch (see also \((^1)\)).

Theorem 5. Let some surface \(\sigma(B_0,B_1)\) divide \(G\) into two parts \(G_1\) and \(G_2\), taken with bases \(B_0\) and \(\sigma\), \(\sigma\) and \(B_1\). Then

\[ \Gamma^{\frac{1}{1-n}}(G_1)+\Gamma^{\frac{1}{1-n}}(G_2)\leq \Gamma^{\frac{1}{1-n}}(G), \]

and equality holds only in the case when on the indicated surface \(u=\mathrm{const}\), where \(u\) is the potential of the extremal function of the domain \(G\).

For the proof it is necessary to estimate \(\Gamma(G_1)\) and \(\Gamma(G_2)\) from below by means of (9), taking \(\rho\) extremal for \(G\), and to use property 3 of Theorem 1.

Theorem 6. Let some piecewise-smooth surface \(\sigma\) divide \(G\) into two parts \(\overline{G}_1\) and \(\overline{G}_2\) and intersect both bases of the domain \(G\), also dividing each of them into two parts, considered as bases for \(\overline{G}_1\) and \(\overline{G}_2\). Then

\[ \Gamma(\overline{G}_1)+\Gamma(\overline{G}_2)\leq \Gamma(G), \]

and equality holds only in the case when on the indicated surface \(\partial u/\partial \nu\equiv 0\), where \(\nu\) is the normal and \(u\) is the potential of the extremal function.

For the proof it is necessary to estimate \(\Gamma(\overline{G}_1)\) and \(\Gamma(\overline{G}_2)\) from above by means of (9), taking \(\rho\) extremal for \(G\), and to use property 2 of Theorem 1.

1. Let \(P^*=f(P)\) be a conformal mapping of \(G\) onto \(G^*\).

Theorem 7. If \(u(P)\) satisfies equation (11), then \(u^*(P^*)=u(P)\) also satisfies this equation.

To verify this, one must, taking \(G\) sufficiently small so as to have (12), use the equalities

\[ h(|\nabla u^*|)=h(|\nabla u|),\quad S(|\nabla u^*|)=S(|\nabla u|),\quad V(|\nabla u^*|)=V(|\nabla u|), \]

equation (13), and the remark to Theorem 4.

We shall denote by an asterisk the image of a set under the transformation of symmetry with respect to an \((n-1)\)-dimensional sphere.

Theorem 8. Let \(B_1\) be a piece of an \((n-1)\)-dimensional sphere. Then, if \(u\) is the potential of the function extremal for \(G\), the function

\[ \rho(P)= \begin{cases} |\nabla u|, & \text{if } P\in G,\\ |\nabla u^*|, & \text{if } P\in G^*,\quad u^*(P^*)=u(P), \end{cases} \]

will be extremal for the domain \(G+G^*\), taken with bases \(B_0\) and \(B_0^*\). Moreover,

\[ \Gamma^{\frac{1}{1-n}}(G+G^*)=2\Gamma^{\frac{1}{1-n}}(G). \tag{14} \]

Indeed, the extremality of \(\rho\) for \(G+G^*\) follows from Theorem 7 and the remark to Theorem 4, while formula (14) follows from Theorem 5.

If, for example, \(G\) is a Grötzsch ring, then \(G+G^*\) will be a Teichmüller ring, and (14) will give a relation between the moduli of these rings, which was obtained by another method in \((^2)\) by Gehring. As the bases of \(G\) one must choose the components of the boundary.

In conclusion I express my gratitude to Prof. B. V. Shabat for valuable advice and suggestions on the subject of this note.

Moscow Forestry Engineering Institute

Received
19 VII 1963

References

\(^1\) B. V. Shabat, DAN, 130, No. 6 (1960).
\(^2\) F. W. Gehring, Trans. Am. Math. Soc., 101, 499 (1961).
\(^3\) G. Federer, Trans. Am. Math. Soc., 93, 426 (1959).
\(^4\) F. W. Gehring, Trans. Am. Math. Soc., 103, 353 (1962).