Abstract
Full Text
UDC 517.54
MATHEMATICS
S. L. KRUSHKAL
ON EXTREMAL QUASICONFORMAL MAPPINGS WITH A PRESCRIBED BOUNDARY CORRESPONDENCE
(Presented by Academician M. A. Lavrent'ev, 5 X 1966)
- A quasiconformal homeomorphism \(w=f(z)\) of the disk \(U:\ |z|<1\) onto the disk \(U':\ |w|<1\) with (measurable) characteristic \(\mu(z)=w_{\bar z}/w_z\) (related to the characteristics \(p(z), \theta(z)\) of M. A. Lavrent'ev \((^1)\) by the relation \(\mu=-[(p-1)/(p+1)]e^{2i\theta}\)), for which
\[ \operatorname*{vrai\,max}_{z\in U}|\mu(z)|=k<1, \]
i.e., the maximal dilatation \(K[f]=(1+k)/(1-k)<\infty\), induces a homeomorphism \(\tau=\omega(t)\) of the circle \(S:\ |t|=1\) onto \(S':\ |\tau|=1\), satisfying the known conditions (see \((^2)\)). Let \(\Omega\) be the set of all homeomorphisms \(\tau=\omega(t): S\to S'\) admitting a quasiconformal extension into the disk \(U\), and let \(Q(\omega)\) be the class of all quasiconformal mappings \(w=f(z): U\to U'\) with the restriction \(f|_S=\omega\), for a given \(\omega\in\Omega\). In this note we consider properties of mappings for which
\[ K[f_0]=\inf_{f\in Q(\omega)} K[f]. \]
It is known \((^{3-6})\) that in the class of quasiconformal homeomorphisms \(w=f(z)\) of the disk \(U\) onto \(U'\), carrying prescribed boundary points \(z_1,\ldots,z_n,\ |z_i|=1\), to prescribed points \(w_1,\ldots,w_n,\ w_i=f(z_i)\), there exists, and moreover uniquely, a mapping with least maximal dilatation \(K[f]\). This extremal mapping has a characteristic of the form \(\mu(z)=k\varphi(z)/|\varphi(z)|\), where \(k=\mathrm{const}\), and \(\varphi(z)\) is a function analytic in the disk \(|z|\leq 1\), except, possibly, for the points \(z_1,\ldots,z_n\), where it may have simple poles*. If, however, one considers quasiconformal mappings with a prescribed boundary correspondence \(\omega\), then in this class there may exist extremal mappings which no longer possess analogous properties. Indeed, as an example constructed in \((^7)\), no. 7, shows, there exist homeomorphisms \(\omega\in\Omega\) such that in the class \(Q(\omega)\) there are, moreover distinct, extremal mappings for which \(|\mu(z)|\ne \mathrm{const}\), as well as such mappings that \(|\mu(z)|=k=\mathrm{const}\), but \(\mu(z)\) cannot be represented in the form \(\mu(z)=k\overline{\varphi(z)}/|\varphi(z)|,\ \varphi\in A(U)\), where \(A(U)\) denotes the Banach space of analytic functions in the disk \(U\) with finite norm
\[
\|\varphi\|_{A(U)}=\iint_U |\varphi(z)|\,dx\,dy,\quad z=x+iy.
\]
A quasiconformal mapping \(w=f_0(z): U\to U'\) with characteristic of the form
\[
\mu(z)=k\overline{\varphi(z)}/|\varphi(z)|,\quad k=\mathrm{const},\quad \varphi\in A(U),
\tag{1}
\]
will be called a Teichmüller mapping. In \((^8)\) it is proved that the Teichmüller mapping \(f_0(z)\) is the unique extremal mapping in the class \(Q(\omega)\), where \(\omega=f_0|_S\).
- For a given \(\omega\in\Omega\), put
\[ K_0[\omega]=\inf_{f\in Q(\omega)} K[f] \]
and
\[ k_0=(K_0[\omega]-1)/(K_0[\omega]+1). \]
Denote by \(f_n(z)\) the extremal with respect to
* In the case when the parameter \(z\) is local, the quadratic differential \(\varphi dz^2\) is an invariant.
the Teichmüller mapping in the class of all quasiconformal homeomorphisms \(f: U \to U'\) satisfying the conditions
\(f(e^{i\pi l/2^{n-1}})=\omega(e^{i\pi l/2^{n-1}})\), \(n=1,2,\ldots;\ l=0,1,\ldots,2^n-1\). We shall denote the characteristic of the mapping \(f_n(z)\) by \(\mu_n(z)\) and put \(|\mu_n(z)|=k_n\). Then \(k_{n+1}\ge k_n\). Mappings that are limits of subsequences of \(\{f_n\}\) (with respect to uniform convergence in the disk \(|z|\le 1\)) will be called limit mappings for the sequence \(\{f_n\}\).
Every limit mapping \(w=f_0(z)\) has maximal dilatation \(K[f_0]=K_0[\omega]\), and, consequently,
\[ \lim_{n\to\infty} k_n=k_0. \tag{2} \]
Lemma. Whatever the sequence of positive numbers \(\{k_n\}\) satisfying the conditions \(k_{n+1}\ge k_n\) and \(\lim_{n\to\infty}k_n=k_0<1\), there exists a homeomorphism \(\omega\in\Omega\) for which \(|\mu_n(z)|=k_n\), \(n=1,2,\ldots\), and \(K_0[\omega]=(1+k_0)/(1-k_0)\).
Proof. Assuming, without loss of generality, that \(k_1=0\) and \(f_1(z)=z\), choose a point \(\tau\), \(|\tau|=1\), so that the conformal quadrilateral with vertices at the points \(1,i,-1,\tau\), formed by the circle \(|w|=1\), has modulus equal to \((1+k_2)/(1-k_2)\). As \(f_2(z)\) we take the mapping, extremal in the sense of Teichmüller, in the class of homeomorphisms \(f: U\to U'\) satisfying the conditions \(f(e^{i\pi l/2})=e^{i\pi l/2}\), \(l=0,1,2\); \(f(-i)=\tau\). Assuming that the mapping \(f_n(z)\) has already been constructed, we construct \(f_{n+1}(z)\) as follows. If \(f_{n+1}^*(z)\) is the extremal mapping in the class of homeomorphisms \(f: U\to U'\) for which \(f(e^{i\pi l/2^n})=f_n(e^{i\pi l/2^n})\), \(l=0,1,\ldots,2^n-2\), \(f(e^{i\pi(2^n-1)/2^n})=\tau_n\), where \(\tau_n\) varies continuously on the circle \(|w|=1\) from the point \(w=f_n(e^{i\pi(2^n-1)/2^n})\) to \(w=1\), then \(K[f_{n+1}^*]\) increases continuously from \(K[f_n]\) to \(\infty\). Therefore there exists such a \(\tau_n=\tau_n'\) that
\(K[f_{n+1}^*]=(1+k_{n+1})/(1-k_{n+1})\), and we put \(f_{n+1}(z)=f_{n+1}^*(z)\) for \(\tau_n=\tau_n'\). In this case the induced homeomorphisms \(\omega_n(t)=f_n|_s\) converge uniformly to some homeomorphism \(\omega\in\Omega\), and \(K_0[\omega]=(1+k_0)/(1-k_0)\). The lemma is proved.
3. Theorem. Let \(\omega\in\Omega\) and \(k_0-k_n=O(2^{-n(1+\alpha)})\), \(\alpha>0\). If there exists a limit mapping \(f_0(z)\) for the sequence \(\{f_n\}\) with characteristic \(\mu_0(z)\), for which \(|\mu_0(z)|=k_0\) almost everywhere in \(U\), then
\[ \sup_{\varphi\in A(U),\,\|\varphi\|_{A(U)}=1} \left|\iint_U \mu_0(z)\varphi(z)\,dx\,dy\right|=k_0. \tag{3} \]
The proof of this theorem is based on the method developed in \((^6)\), and proceeds according to the following scheme.
Suppose there exists a limit mapping \(w=f_0(z)\) for the sequence \(\{f_n\}\) with characteristic \(\mu_0(z)\), for which \(|\mu_0(z)|=k_0\) almost everywhere in \(U\), and let \(\{f_{n_i}(z)\}\subset\{f_n(z)\}\) be a subsequence of mappings (which, for convenience, we again denote by \(\{f_n\}\)) converging to \(f_0(z)\) uniformly in the disk \(|z|\le 1\), with \(f_0(z)\ne f_n(z)\), \(n=1,2,\ldots\).
In the space \(L_1(U)\) of functions \(\varphi(z)\) measurable in \(U\), with norm
\(\|\varphi\|_{L_1(U)}=\iint_U|\varphi(z)|\,dx\,dy\), consider the functional
\(\mu_0(\varphi)=\iint_U\mu_0(z)\varphi(z)\,dx\,dy\). Let
\[ \sup_{\varphi\in A(U),\,\|\varphi\|_{A(U)}=1} \left|\iint_U \mu_0(z)\varphi(z)\,dx\,dy\right|=k'. \tag{3'} \]
We shall show that \(k'=k_0\). Suppose that \(k'<k_0\). Then, by the Hahn–Banach theorem, in \(L_1(U)\) there is a linear functional \(m_0(\varphi)\) such that \(m_0(\varphi)=\mu_0(\varphi)\) for \(\varphi\in A(U)\) and \(\|m_0\|_{L_1(U)}=k'\). In this case
\[ m_0(\varphi)=\iint_U m_0(z)\varphi(z)\,dx\,dy, \tag{4} \]
where \(m_0(z)\) is a measurable function in the disk \(U\) and \(\operatorname*{vrai\,max}_{z\in U}|m_0(z)|=k'\). Then for the difference \(v(z)=\mu_0(z)-m_0(z)\) we shall have
\[ \iint_U v(z)\varphi(z)\,dx\,dy=0,\qquad \varphi\in A(U). \tag{5} \]
The function \(v(z)\) determines the corresponding variation of the disk, i.e. the mapping \(\zeta=H(z,\varepsilon)\) of the disk \(U\) onto itself with characteristic \(\varepsilon v(z)\) and normalization \(H(1,\varepsilon)=1,\ H(i,\varepsilon)=i,\ H(-1,\varepsilon)=-1\), representable by the formula
\[ \zeta=H(z,\varepsilon)=z-\frac{\varepsilon}{\pi}\iint_U \left[\frac{v(\xi)}{\xi-z}+\frac{z^3\overline{v(\xi)}}{1-z\bar\xi}\right]\,d\sigma(\xi)+M(z)+O(\varepsilon^2), \quad |z|\leq 1, \tag{6} \]
where \(\varepsilon\) is a small real parameter, \(M(z)\) is a polynomial of the form \(M(z)=a+2ibz-\bar a z^2\), whose coefficients are uniquely determined from the normalization conditions (see (9)).
Denoting the characteristic of the mapping \(w=f_0\circ H^{-1}(\zeta)\) by \(\mu^*(\zeta)\), we obtain:
\[ \tilde\mu(z)\equiv \mu^*(\zeta(z))\,\overline{\zeta_z}/\zeta_z =\mu_0(z)-\varepsilon v(z)+\varepsilon\overline{v(z)}\mu_0^2(z)+O(\varepsilon^2). \tag{7} \]
By applying equalities (4) and (7) it is established that, for sufficiently small \(\varepsilon>0\), the inequality
\[ \sup_{\|\varphi\|_{L_1(U)}=1} \left|\iint_U \tilde\mu(z)\varphi(z)\,dx\,dy\right| <k_0-O(\varepsilon) \tag{8} \]
holds, where the quantity \(O(\varepsilon)\) depends only on \(\varepsilon\), \(k_0\), and \(k'\), i.e.
\[ k^*=\operatorname*{vrai\,max}_{\zeta\in U}|\mu^*(\zeta)|<k_0-O(\varepsilon). \tag{9} \]
On the other hand, by virtue of (6),
\[ \zeta_{l,n}\equiv H\left(e^{i\pi l/2^{\,n-1}},\varepsilon\right) =e^{i\pi l/2^{\,n-1}}+O(\varepsilon^2), \qquad n=1,2,\ldots;\ l=0,1,\ldots,2^n-1, \]
and, consequently, denoting by \(\tilde\mu_n(\zeta)\) the characteristic of the Teichmüller extremal mapping \(\tilde f_n(\zeta)\) in the class of homeomorphisms \(f:U\to U'\) satisfying the conditions
\(f(\zeta_{l,n})=\omega\circ H^{-1}(\zeta_{l,n})\), \(l=0,1,\ldots,2^n-1\), for fixed \(n\), we shall have
\[ |\tilde k_n-k_n|<O(2^n\varepsilon^2), \qquad \text{where } \tilde k_n=|\tilde\mu_n(\zeta)|. \tag{10} \]
In particular, if we take a sufficiently large \(n=n_0\) and \(\varepsilon=2^{-n(1+\alpha/4)}\), then from (9) and (10) we obtain the inequality \(k^*<|\mu_{n_0}(\zeta)|\), which is impossible. The contradiction obtained proves that \(k'=k_0\), i.e. (3) is satisfied.
If the conditions are satisfied that ensure the existence of an element \(\varphi_0\in A(U)\), \(\|\varphi_0\|_{A(U)}=1\), for which \(\sup|\mu_0(\varphi)|\) is attained on the sphere \(\|\varphi\|_{A(U)}=1\) (which certainly occurs in the case when only a finite number of points are fixed on the circles \(S\) and \(S'\)), then from (3) it follows that \(\mu_0(z)=k_0\varphi_0(z)/|\varphi_0(z)|\), i.e. the mapping \(f_0(z)\) is Teichmüller and, consequently, the unique extremal one in the class \(Q(\omega)\).
Institute of Mathematics
Siberian Branch of the Academy of Sciences of the USSR
Received
20 IX 1966
REFERENCES
- M. A. Lavrent’ev, Variational method in boundary-value problems for systems of elliptic-type equations, Moscow, 1962.
- A. Beurling, L. V. Ahlfors, Acta math., 96, 125 (1956).
- O. Teichmüller, Abh. Preuss. Akad. Wiss., Math.-Naturwiss. Kl., 22, 1 (1939).
- L. Ahlfors, in: L. Ahlfors and L. Bers, Riemann surface structure and quasiconformal mappings, IL, 1961, p. 104.
- L. Bers, ibid., p. 9.
- S. L. Krushkal’, DAN, 171, No. 4 (1966).
- K. Strebel, Comm. Math. Helv., 36, No. 4, 306 (1962).
- K. Strebel, Comm. Math. Helv., 39, No. 2—3, §1 (1964).
- L. V. Ahlfors, Ann. Math., 74, No. 1, 171 (1961).