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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
##### MSC:
 30C70 Extremal problems for conformal and quasiconformal mappings, variational methods 35J55 Systems of elliptic equations, boundary value problems (MSC2000)
##### Keywords:
quasilinear Beltrami equation