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On a variational method for quasiconformal mappings. (English. Russian original) Zbl 0603.30027
Sov. Math., Dokl. 32, 614-617 (1985); translation from Dokl. Akad. Nauk SSSR 285, 24-27 (1985).
Authors develop a variational method for qc maps \(f: {\mathbb{C}}\to {\mathbb{C}}\) with the hydrodynamical normalization at \(\infty\) which satisfy the quasilinear Beltrami equation \(f_{\bar z}=\nu (z,f)f_ z\), where \[ | \nu (z,w)| \leq q(z,w)\leq q<1,\quad \nu (z,w): {\mathbb{C}}^ 2\to {\mathbb{C}}\quad and\quad q(z,w): {\mathbb{C}}^ 2\to [0,q] \] verify the Carathéodory conditions (they are measurable in z for \(w\in {\mathbb{C}}\) and continuous in w for a.e. \(z\in {\mathbb{C}})\) and supp q(\(\cdot,w)\subset U=\{| z| <1\}\) for \(w\in {\mathbb{C}}\). If \[ Q(z,w)=(1+q(z,w))(1- q(z,w))^{-1}, \] let \(\Sigma_{Q(z,w)}\) denote the class of all these maps f when \(\nu\) (z,w) varies; \(\Sigma_{Q(z,w)}\subset \Sigma_ Q\) and it is sequentially compact with respect to locally uniform convergence. Further suppose that: \(q(z,w)\geq q_ 0>0\) for \(z\in U\), \[ q(z+w+\Delta w)=q(z,w)+2 Re(q_ w(z,w)\Delta w)+o(\Delta w), \] \(q_ w(z,w)\) verifies the Carathéodory conditions and \(| q_ w(z,w)| \leq C<\infty\). Let \(\mu\) be the complex characteristic of f, \(\sigma\) a function of \(L^{\infty}(U)\) such that ess inf Re \(\sigma\) (z)\(>0\) on \[ E(\delta)=\{z\in U: | \mu (z)| >q(z,f)-\delta \},\quad 0<\delta <q_ 0, \] and \[ T(g)(w)=(2\pi i)^{- 1}\iint_{{\mathbb{C}}}g(\zeta)(\zeta -w)^{-1} d\zeta d{\bar \zeta} \] for every \(g\in L^ 1_ 0\). The authors prove the existence of a variation \(f_ t\) of f in \(\Sigma_{Q(z,w)}\) of the form \(f_ t=f+t\Phi \circ f+o(t)\), where \(\Phi\) (w) is the unique generalized solution of the equation \(\Phi_{\bar w}=h(w)\Phi +k(w)\) with \(T(\Phi_{\bar w})=\Phi\), h and k being given by formulae depending on f, q(z,f), \(q_ w(z,f)\), \(\mu\), \(\gamma (z)=(1-| \mu (z)|^ 2)^{-1} f_ z/\overline{f_ z}\) and \(\sigma\). Namely \(\Phi (w)=\phi (w)T(k/\phi)\), \(\phi =\exp T(h)\). Under some hypotheses, necessary conditions for the complex characteristic of the extremal map \(f\in \Sigma_{Q(z,w)}\) which maximizes an upper semicontinuous functional on \(\Sigma_{Q(z,w)}\) are established.
Reviewer: C.Andreian Cazacu
30C70 Extremal problems for conformal and quasiconformal mappings, variational methods
35J55 Systems of elliptic equations, boundary value problems (MSC2000)