Abstract
Full Text
V. V. KRIVOV
ON EXTREMAL QUASICONFORMAL MAPPINGS IN SPACE
(Presented by Academician M. A. Lavrent′ev on 28 II 1962)
In the theory of \(Q\)-quasiconformal mappings of plane domains, extremal mappings (with the least possible \(Q\)) have been well studied. In this note we consider, in a certain sense, extremal mappings of some spatial domains. The proofs are carried out by the method of moduli (see \((^1)\)) in space.
Let us introduce the following notation. By \(D\) we shall denote the cylindrical domain of three-dimensional Euclidean space \((x,y,z)\), \((x,y)\in G\), \(0\le z\le H\), where \(G\) is a bounded closed domain in the \((x,y)\)-plane. If some smooth non-self-intersecting surface passes inside \(D\) and intersects its bases, then it divides \(D\) into two parts, denoted by \(D_1\) and \(D_2\).
By \(\{C\}\) we denote the family of all rectifiable curves lying inside \(D\) and joining its bases; into the family \(\{C_1\}\) we include the curves from \(\{C\}\) that lie entirely inside \(D_1\), and into \(\{C_2\}\) the curves from \(\{C\}\) that lie entirely inside \(D_2\). \(M\{C\}\) is the modulus of the family \(\{C\}\). The cylindroids \((x,y)\in G\), \(0\le z\le f(x,y)\), and \((x,y)\in G\), \(f(x,y)\le z\le H\) will be denoted respectively by \(\overline{D}_1\) and \(\overline{D}_2\). It is assumed that \(f(x,y)\) is continuous and \(0<f(x,y)<H\) everywhere in the domain \(G\). The family of all rectifiable curves lying inside \(\overline{D}_1\) (respectively inside \(\overline{D}_2\)) and joining its bases will be denoted by \(\{\overline{C}_1\}\) (respectively \(\{\overline{C}_2\}\)).
- Lemma 1. In the relation
\[ M\{C_1\}+M\{C_2\}\le M\{C\} \tag{1} \]
equality is attained only for a cylindrical surface with generators parallel to the \(z\)-axis.
Indeed, otherwise one can find on our surface such a point \(P_0\) that, for any curve from \(\{C_1\}\) (or perhaps from \(\{C_2\}\)) entering a certain \(\varepsilon\)-neighborhood \(V_\varepsilon(P_0)\) of the point \(P_0\), the length of the part of it lying outside \(V_\varepsilon(P_0)\) will be greater than \(H+2\varepsilon\). Then the metric
\[
\rho_1(P)=
\begin{cases}
\dfrac{1}{H}, & \text{if } P\in D_1-V_\varepsilon(P_0),\\[6pt]
0, & \text{if } P\in D_1\cap V_\varepsilon(P_0),
\end{cases}
\]
will be admissible for \(\{C_1\}\), while \(\rho_2(P)=\dfrac{1}{H}\), \(P\in D_2\), is admissible for \(\{C_2\}\). Therefore
\[
M\{C_1\}+M\{C_2\}
\le
\iiint_{D_1}\rho_1^3\,dV+\iiint_{D_2}\rho_2^3\,dV
<
\iiint_D \frac{1}{H^3}\,dV
=
M\{C\}.
\]
The assertion concerning equality is obvious.
Lemma 2. In the relation
\[
\frac{1}{\sqrt{M\{\overline{C}_1\}}}
+
\frac{1}{\sqrt{M\{\overline{C}_2\}}}
\le
\frac{1}{\sqrt{M\{C\}}}
\tag{2}
\]
equality is attained only in the case \(f(x,y)\equiv \mathrm{const}\).
Dividing \(G\) into parts \(G_i\) \((i=1,2,\ldots,n)\) and considering the moduli of families of curves lying in the resulting cylinders \((x,y)\in G_i\), \(0<z<f(x,y)\); \((x,y)\in G_i\), \(f(x,y)<z<H\), by Theorem 4 of \({}^{(1)}\), with the subsequent passage to the limit, we obtain the estimates
\[ \iint_G \frac{d\sigma}{f^2(x,y)} \leqslant M\{\overline{C}_1\}, \qquad \iint_G \frac{d\sigma}{[H-f(x,y)]^2} \leqslant M\{\overline{C}_2\}. \]
In conjunction with the known inequality from \({}^{(2)}\)
\[ \left(\frac{1}{S}\iint_G f^{-2}(x,y)\,d\sigma\right)^{-1/2} < \frac{1}{S}\iint_G f(x,y)\,d\sigma, \]
where \(S=\iint_G d\sigma\) and \(f(x,y)\not\equiv \mathrm{const}\), these estimates give the required result. The case \(f(x,y)\equiv \mathrm{const}\) is trivial.
- It next turns out to be expedient to introduce the following definition, proposed by B. V. Shabat.
Definition. A homeomorphic mapping \(P_*=f(P)\) of a domain \(D\) onto \(D^*\) is called \((Q_1,Q_2)\)-quasiconformal if it has continuous partial derivatives in \(D\), its Jacobian is everywhere positive, and at each point \(P\in D\) its principal linear part transforms a sphere into an ellipsoid in which the ratio of the major axis to the minor axis is bounded above by the quantity \(Q_1\), and the ratio of the intermediate axis to the minor axis by the quantity \(Q_2\). The inverse of a \((Q_1,Q_2)\)-quasiconformal mapping is, obviously, also \((Q_1,Q_2)\)-quasiconformal with the same \(Q_1\) and \(Q_2\).
Theorem 1. If \(\{C^*\}\) is the image of the family of curves \(\{C\}\) under a \((Q_1,Q_2)\)-quasiconformal mapping, then
\[ \frac{M\{C\}}{Q_1Q_2}\leqslant M\{C^*\}\leqslant Q_1Q_2M\{C\}. \tag{3} \]
The proof is almost the same as that of Theorem 1 from \({}^{(1)}\).
- Alongside \(D\), consider in the space \((u,v,w)\) the cylinder \(D^*\), \((u,v)\in G^*\), \(0<w<h\). We shall assume that \(D^*\) can be obtained from \(D\) by the affine transformation
\[ u=\lambda_1 x,\qquad v=\lambda_2 y,\qquad w=\frac{h}{H}z, \tag{4} \]
where \(\lambda_1\geqslant \lambda_2\geqslant h/H\). The notation \(\{C^*\}\), \(\{C_1^*\}\), \(D_1^*\), etc., plays for \(D^*\) the same role as the corresponding notation without asterisks for \(D\).
Let \(\mathfrak{M}\) denote the class of \((Q_1,Q_2)\)-quasiconformal mappings of \(D\) onto \(D^*\) that take the bases of \(D\) onto the bases of \(D^*\).
Theorem 2. A mapping from the class \(\mathfrak{M}\) possessing the minimal value of the product \(Q_1Q_2\) in comparison with the other mappings of the same class must necessarily have the form
\[ u=u(x,y),\qquad v=v(x,y),\qquad w=\frac{h}{H}z. \tag{5} \]
Let \(\Sigma^*\) be an arbitrary cylindrical surface with generator along the \(w\)-axis, separating \(D^*\) into \(D_1^*\) and \(D_2^*\). If one assumes that its preimage \(\Sigma\) under a mapping from the class \(\mathfrak{M}\) is not a cylindrical surface with generator along the \(z\)-axis, then, by Lemma 1, taking into account inequalities (3), we obtain for such a mapping
\[ M\{C\}>M\{C_1\}+M\{C_2\}\geqslant \frac{M\{C_1^*\}}{Q_1Q_2}+\frac{M\{C_2^*\}}{Q_1Q_2} = \frac{M\{C^*\}}{Q_1Q_2}, \]
or
\[ Q_1Q_2>\frac{M\{C^*\}}{M\{C\}}. \]
The mapping (4) belongs to the class \(\mathfrak M\), and for it
\[ Q_1 Q_2=\frac{M\{C^*\}}{M\{C\}}. \]
Consequently, for the extremal mapping (minimizing \(Q_1Q_2\)), \(\Sigma\) is a cylindrical surface with generators along the \(z\)-axis. Hence it is easy to obtain that any segment parallel to the \(z\)-axis is transformed under the extremal mapping into a segment parallel to the \(w\)-axis.
Thus, for the extremal mapping, the image of the plane \(z=z_0\) can only be a surface \(w=w_0(u,v)\).
Suppose that \(w_0(u,v)\not\equiv \mathrm{const}\). Then Lemma 2 and inequalities (3) give
\[ \frac{1}{\sqrt{M\{C\}}} = \frac{1}{\sqrt{M\{C_1\}}} + \frac{1}{\sqrt{M\{C_2\}}} \leq \frac{\sqrt{Q_1Q_2}}{\sqrt{M\{C_2^*\}}} + \frac{\sqrt{Q_1Q_2}}{\sqrt{M\{C_2^*\}}} < \sqrt{Q_1Q_2}\,\frac{1}{\sqrt{M\{C^*\}}}, \]
whence
\[ Q_1Q_2>\frac{M\{C^*\}}{M\{C\}}, \]
which contradicts the extremality of the mapping.
The linearity
\[ w=\frac{h}{H}\,z \]
is easily proved by contradiction.
Theorem 3. The extremal mapping has constant Jacobian.
This assertion follows easily from the relations:
\[ S^*h=\iiint_D J\,dV = \frac{h}{H}\iiint_D \left| \begin{matrix} u_x & u_y\\ v_x & v_y \end{matrix} \right|\,dV \leq \frac{h^3}{H^3}\,S H Q_1Q_2, \]
where
\[ S^*=\iint_{G^*} d\sigma,\qquad J=\frac{D(u,v,w)}{D(x,y,z)}. \]
Remark. For the extremal mapping (5), the corresponding plane mapping of \(G\) onto \(G^*\),
\[ u=u(x,y),\qquad v=v(x,y) \tag{6} \]
will be \(Q\)-quasiconformal, with
\[ Q=\frac{Q_1^2}{J}\,\frac{h^3}{H^3}. \]
Let us apply to the mapping (6) the theorem on the extremal quasiconformal mapping in the plane (3).
Theorem 4. Among all extremal mappings of the parallelepiped \(0\leq x\leq a,\ 0\leq y\leq b,\ 0\leq z\leq H\) onto the parallelepiped \(0\leq u\leq \lambda_1a,\ 0\leq v\leq \lambda_2b,\ 0\leq w\leq h\) with corresponding edges, the affine mapping is the unique one having the minimal value of \(Q_1\).
Received
8 II 1962
References Cited
- B. V. Shabat, Dokl. Akad. Nauk SSSR, 130, No. 6 (1960).
- G. G. Hardy, J. E. Littlewood, I. G. Polya, Inequalities, Moscow, 1948.
- L. A. Ahlfors, J. Analyse Math., 3, 1 (1954).