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On quasiconformal mappings with restrictions of integral type on the Lavrent’ev characteristic. (English. Russian original) Zbl 0663.30015
Sov. Math., Dokl. 36, No. 3, 456-459 (1988); translation from Dokl. Akad. Nauk SSSR 297, 283-286 (1987).
The authors study classes \({\mathcal F}^{\Phi}\) of quasiconformal mappings f of plane domains defined by conditions of the type \[ \iint_{E}\Phi (p(z),z) dx dy\leq M, \] where \(p(z)=(1+| \mu (z)|)/(1-| \mu (z)|)\) and \(\mu\) is the complex dilatation of f. Assuming some regularity of \(\Phi\) they state convergence and compactness results and give a variational formula by which properties of extremals for certain functionals \(\Omega\) in \({\mathcal F}^{\Phi}\) can be described.
Reviewer: M.Lehtinen
30C62 Quasiconformal mappings in the complex plane
30C70 Extremal problems for conformal and quasiconformal mappings, variational methods