MATHEMATICS

V. V. KRIVOV

BEST EXTREMAL MAPPINGS IN SPACE

(Presented by Academician M. A. Lavrent'ev, November 4, 1963)

In this note we investigate quasiconformal mappings of domains of \(n\)-dimensional space that are closest to conformal ones.

1. Let \(P^* = f(P)\) be a homeomorphic mapping of a simply connected domain \(G\) onto a domain \(G^*\). If for any family of curves \(\{C\} \subset G\) and its image \(\{C^*\} \subset G^*\) we have

\[ \frac{1}{K} M\{C\} \leqslant M\{C^*\} \leqslant K M\{C\}, \tag{1} \]

then the mapping is called quasiconformal \(\left({}^{1}\right)\) (\(M\{C\}\) is the modulus of the family \(\{C\}\)). As \(K\) in (1) it is natural to take the least of the suitable constants.

Mark on the boundary of \(G\) two continua \(B_0\) and \(B_1\), and let \(B_0^*\) and \(B_1^*\) be their images* under the mapping under consideration. Instead of the modulus of the family of curves joining \(B_0\) and \(B_1\) in \(G\), it will be more convenient for us to consider the conformal capacity \(\Gamma(G)\) of the domain \(G\), equal to it \(\left({}^{2,3}\right)\). We have

\[ \frac{1}{K} \leqslant \frac{\Gamma(G^*)}{\Gamma(G)} \leqslant K. \]

Following Väisälä \(\left({}^{1}\right)\), one may write

\[ \inf \frac{J}{\Lambda^n} \leqslant \frac{\Gamma(G^*)}{\Gamma(G)} \leqslant \sup \frac{J}{\lambda^n}, \tag{2} \]

where \(\lambda\) is the least, \(\Lambda\) the greatest stretching at the given point, and \(J\) is the Jacobian of the mapping.

Definition 1. A quasiconformal mapping is called best extremal if

\[ K = \frac{\Gamma(G^*)}{\Gamma(G)} \quad \text{when } \Gamma(G^*) \geqslant \Gamma(G) \]

or

\[ K = \frac{\Gamma(G)}{\Gamma(G^*)} \quad \text{when } \Gamma(G^*) \leqslant \Gamma(G). \]

In other words, a mapping is called best extremal if at least one of the inequalities (2) becomes an equality.

Below the properties of best extremal mappings are clarified.

1. Let \(u\) and \(u^*\) be, respectively, the potentials of the extremal metrics \(\left({}^{3}\right)\) of the domains \(G\) and \(G^*\).

Introduce the mappings:

I. \(P = \tau(Q)\), which sends any point \(Q\) on the level surface \(\Sigma(P)\) of the function \(u\) (so that \(u(Q) = a\)) to the fixed point \(P \in \Sigma(P)\).

II. \(P = \varphi(Q)\), which sends any point \(Q\) on the force line \(L(P)\) of the vector field \(\nabla u\) to the fixed point \(P \in L(P)\).

* The mapping can be extended to the closures of \(G\) and \(G^*\).

We introduce analogous mappings \(P^*=\tau^*(Q^*)\) and \(P^*=\varphi^*(Q^*)\) in the domain \(G^*\). Further, let \(P^*=T(P)\) be a homeomorphic mapping of \(\Sigma(P_0)\) onto \(\Sigma(P_0^*)\) such that \(P_0^*=T(P_0)\), and let \(P^*=F(P)\) be a homeomorphic mapping of \(L(P_0)\) onto \(L(P_0^*)\) such that \(P_0^*=F(P_0)\); \(P_0\) and \(P_0^*\) are fixed.

Definition 2. A mapping \(Q^*=U(Q)\) of the domain \(G\) onto the domain \(G^*\) having the properties: 1) if \(\varphi(Q)=P_1,\ P_1\in\Sigma(P_0)\), and \(T(P_1)=P_1^*\), then \(Q^*\in{\varphi^*}^{-1}(P_1^*)\); 2) if \(\tau(Q)=P_2,\ P_2\in L(P_0)\), and \(F(P_2)=P_2^*\), then \(Q^*\in{\tau^*}^{-1}(P_2^*)\), is called a special mapping.

Consequently, in the case of a special mapping, the image of any level surface \(u=\mathrm{const}\) will be some level surface \(u^*=\mathrm{const}\), and the image of any force line of the field \(\nabla u\) will be some force line of the field \(\nabla u^*\).

If the domains \(G\) and \(G^*\) are simply covered by level surfaces and their orthogonal trajectories, then the special mapping is one-to-one and is completely determined by specifying the mappings \(T\) and \(F\).

Theorem 1. A best extremal mapping is necessarily a special mapping.

The proof of this theorem is based on the application of the Grötzsch principles in the form in which they are given in \((^3)\), and in essence does not differ from the proof of Theorem 2 from \((^4)\).

1. Let us now find out in what case a special mapping will be extremal. Put \(\rho=|\nabla u|\), \(\rho^*=|\nabla u^*|\). Let, for definiteness, \(\Gamma(G^*)\geqslant \Gamma(G)\).

Theorem 2. In order that a special mapping be the best extremal mapping, it is necessary and sufficient that the following conditions be satisfied simultaneously:

1) \(\displaystyle \lambda=\lim_{Q\to P}\frac{|U(Q)-U(P)|}{|Q-P|},\) where \(Q\in\varphi^{-1}(P)\);

2) \(l'(\alpha)=\mathrm{const}\), if we put \(u(Q)=\alpha,\ u^*(Q^*)=l(\alpha)\), if \(Q^*=U(Q)\);

3) \(\displaystyle \int J/\lambda^n=\mathrm{const}.\)

For the proof, consider the function

\[ \mu(P)=\lim_{Q\to P}\frac{|U(Q)-U(P)|}{|Q-P|},\qquad \text{where } Q\in\varphi^{-1}(P). \]

We shall show that

\[ \frac{\Gamma(G^*)}{\Gamma(G)}\leqslant \sup_{P\in G}\frac{J}{\mu^n}. \tag{3} \]

Putting

\[ h(\rho)=\int_\gamma \rho\,ds,\quad \text{if } \gamma=\varphi^{-1}(P), \]

we obtain

\[ h\left[\left(\frac{\rho}{\mu}\right)^*\right] = \int_{\gamma^*}\left(\frac{\rho}{\mu}\right)^*ds^* = \int_\gamma \frac{\rho}{\mu}\frac{ds^*}{ds}\,ds = \int_\gamma \rho\,ds = h(\rho), \tag{4} \]

where \(\left(\dfrac{\rho}{\mu}\right)^*=\dfrac{\rho}{\mu}\), if \(P^*=U(P)\).

Then from the inequality

\[ \Gamma(G^*)\leqslant \frac{1}{h^n\left[(\rho/\mu)^*\right]} \int_{G^*}\left[\left(\frac{\rho}{\mu}\right)^*\right]^n dV = \frac{1}{h^n(\rho)} \int_G \rho^n\frac{J}{\mu^n}\,dV \tag{5} \]

we shall have

\[ \Gamma(G^*) \leq \frac{1}{h^n(\rho)} \int_G \rho^n \frac{J}{\mu^n}\, dV \leq \sup \frac{J}{\mu^n}\cdot \frac{1}{h^n(\rho)} \int_G \rho^n\, dV = \Gamma(G)\sup \frac{J}{\mu^n}, \tag{6} \]

which proves (3). Since \(\lambda \leq \mu\), for an extremal mapping

\[ \frac{\Gamma(G^*)}{\Gamma(G)}=\sup \frac{J}{\mu^n}=\sup \frac{J}{\lambda^n}. \tag{7} \]

Consequently, for the best extremality it is necessary and sufficient that the equalities in (5), (6) be realized. In (5) this is possible only in the case where \(\left(\frac{\rho}{\mu}\right)^*\) is an extremal metric for \(G^*\), whence

\[ \left(\frac{\rho}{\mu}\right)^*=\operatorname{const}\cdot \rho^*, \]

so that

\[ \mu^{-1}=\operatorname{const}\cdot \frac{\rho^*(P^*)}{\rho(P)}, \tag{8} \]

from which assertion 2) follows.

In (6) the equalities are attained only if

\[ \frac{J}{\mu^n}\equiv \sup \frac{J}{\mu^n}=\operatorname{const}, \tag{9} \]

so that

\[ \sup \frac{J}{\mu^n}=\frac{J}{\mu^n}\leq \frac{J}{\lambda^n}\leq \sup \frac{J}{\lambda^n}=\sup \frac{J}{\mu^n}, \]

whence property 3) and the identity \(\lambda\equiv\mu\), i.e., property 1), follow. The theorem is proved.

In the case \(\Gamma(G^*)\leq \Gamma(G)\), the formulation of the theorem is unchanged; one need only replace \(\lambda\) by \(\Lambda\), and the equality in (2) will be attained on the left. To verify this, it is enough to apply Theorem 2 to the best extremal mapping in the class of mappings of \(G^*\) onto \(G\).

If in (2) the equalities are attained on both sides at once, then

\[ \frac{J}{\lambda^n}=\frac{J}{\Lambda^n}=\operatorname{const}, \]

so that \(\lambda=\Lambda\), and the mapping will be conformal.

Remark. If properties 1) and 2) hold in \(G\), and property 3) holds for some level surface \(u=\alpha\), then this property holds everywhere in \(G\).

This may be verified by considering the vector tubes of the fields \(\nabla u\) and \(\nabla u^*\) and using the fact that the integral \(\int_{d\sigma(\alpha)} |\nabla u|^{\,n-1}\, dS\), where \(d\sigma(\alpha)\) is the section of the vector tube by a piece of the level surface \(u=\alpha\), does not depend on \(\alpha\) \((^3)\).

I express my gratitude to Prof. B. V. Shabat for discussing the work.

Moscow State University
named after M. V. Lomonosov

Received
10 IX 1963

CITED LITERATURE

1. J. Väisälä, Ann. Acad. Sci. Fenn., Ser. A. I., No. 298, 1 (1961).
2. F. W. Gehring, Michigan Math. J., 9 (1962).
3. V. V. Krivov, DAN, 154, No. 3 (1964).
4. V. V. Krivov, DAN, 145, No. 3 (1962).