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A variational method for general families of quasiconformal mappings. (English) Zbl 0419.30018


MSC:

30C70 Extremal problems for conformal and quasiconformal mappings, variational methods
30C62 Quasiconformal mappings in the complex plane
30C50 Coefficient problems for univalent and multivalent functions of one complex variable
30G20 Generalizations of Bers and Vekua type (pseudoanalytic, \(p\)-analytic, etc.)
35J50 Variational methods for elliptic systems
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References:

[1] Bers, L., Univalent solutions of linear elliptic systems, Comm. Pure Appl. Math., 6, 513-526 (1953) · Zbl 0051.31701 · doi:10.1002/cpa.3160060407
[2] Carathéodory, C., Vorlesungen über Reelle Funktionen (1948), New York: Chelsea Publ. Co., New York
[3] Courant, R.; Hilbert, D., Methods of Mathematical Physics, Vol. II (1962), New York: Interscience Publ., New York · Zbl 0099.29504
[4] Garabedian, P. R., Partial Differential Equations (1964), New York: John Wiley and Sons, New York · Zbl 0124.30501
[5] Kühnau, R., Wertannahmeprobleme bei quasikonformen Abbildungen mit ortsabhängiger Dilatationsbeschränkung, Math. Nachr., 40, 1-11 (1969) · Zbl 0195.09201 · doi:10.1002/mana.19690400102
[6] M. A. Lavrent’ev,The general propblem of the theory of quasiconformal mappings of plane regions, Amer. Math. Soc. Transl. No. 46 (1951).
[7] Lehto, O., Schlicht functions with a quasiconformal extension, Ann. Acad. Sci. Fenn. Ser. A, I, 500-500 (1971) · Zbl 0226.30020
[8] Lehto, O.; Virtanen, K. I., Quasiconformal Mappings in the Plane (1973), New York-Heidelberg-Berlin: Springer-Verlag, New York-Heidelberg-Berlin · Zbl 0267.30016
[9] Schiffer, M., A variational method for univalent quasiconformal mappings, Duke Math. J., 33, 395-412 (1966) · Zbl 0152.07303 · doi:10.1215/S0012-7094-66-03345-X
[10] Schiffer, M.; Schober, G., Representation of fundamental solutions for generalized Cauchy-Riemann equations by quasiconformal mappings, Ann. Acad. Sci. Fenn. Ser. A, I, 501-531 (1976) · Zbl 0362.30021
[11] G. Schober,Univalent Functions—Selected Topics, Springer-Verlag Lecture Notes in Math.478, 1975. · Zbl 0306.30018
[12] Strebel, K., Ein Konvergensatz für Folgen quasikonformer Abbildungen, Comment. Math. Helv., 44, 469-475 (1969) · Zbl 0184.31002 · doi:10.1007/BF02564546
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