Abstract
We have constructed variations for classes of regular solutions of the degenerate Beltrami equation with constraints of the set-theoretic and integral types for the complex coefficient and, on this basis, proved the variational principles of maximum and other necessary conditions of extremum. Some applications to equations of mathematical physics are given.
This is a preview of subscription content, log in via an institution to check access.
Similar content being viewed by others
References
L. V. Ahlfors, Lectures on Quasiconformal Mappings, Van Nostrand-Reinhold, Princeton, NJ, 1966. 1969.
L. Ahlfors and L. Bers, “Riemann’s mapping theorem for variable metrics,” Ann. Math., 72, 385–404 (1960).
P. P. Belinskii, General Properties of Quasiconformal Mappings [in Russian], Nauka, Novosibirsk, 1974.
B. Bojarski, V. Gutlyanskii, and V. Ryazanov, “On the Beltrami equations with two characteristics,” Complex Var. Ellipt. Eq., 54, No. 10, 935–950 (2009).
B. Bojarski, V. Gutlyanskii, and V. Ryazanov, “On integral conditions for general Beltrami equations,” Complex Anal. Oper. Theory, 5, No. 3, 835–845 (2011).
N. Bourbaki, Integration, Springer, Berlin, 2004.
G. David, “Solutions de l’equation de Beltrami avec ‖μ‖ ∞ = 1,” Ann. Acad. Sci. Fenn., Ser. A1. Math., 13, 25–70 (1988).
H. Federer, Geometric Measure Theory, Springer, Berlin, 1969.
G. M. Fikhtengol’ts, Course of Differential and Integral Calculus [in Russian], Vol. 1, Nauka, Moscow, 1970.
G. M. Goluzin, Geometric Theory of Functions of a Complex Variable, Amer. Math. Soc., Providence, RI, 1969.
V. M. Gol’dshtein and Yu. G. Reshetnyak, Introduction to the Theory of Functions with Generalized Derivatives and Quasiconformal Mappings [in Russian], Nauka, Moscow, 1983.
V. Ya. Gutlyanskii, “On the method of variations for schlicht analytic functions with a quasiconformal continuation,” Dokl. AN SSSR, 236, No. 5, 1045–1048 (1977).
V. Ya. Gutlyanskii, “On the product of conformal radii of nonoverlapping domains,” Dokl. AN Ukr. SSR, Ser. A, No. 4, 298–302 (1977).
V. Ya. Gutlyanskii, “On the method of variations for schlicht analytic functions with a quasiconformal continuation,” Sib. Mat. Zh., 21, No. 2, 61–78 (1980).
V. Ya. Gutlyanskii, “The product of the conformal radii of nonoverlapping domains,” Amer. Math. Soc. Transl., Ser. 2, 122, 65–69 (1984).
V. Ya. Gutlyanskii and V. I. Ryazanov, “To the theory of a local behavior of quasiconformal mappings,” Izv. RAN, Ser. Mat., 59, No. 3, 31–58 (1995).
V. Ya. Gutlyanskii and V. I. Ryazanov, “On the fundamental solution of an equation of mathematical physics,” in Complex Methods in Mathematical Physics, Abstr. of Reports, All-Union School of Young Scientists [in Russian], IM AN SSSR, IPMM AN Ukr. SSR, DGY, Donetsk, 1984.
V. Ya. Gutlyanskii and V. I. Ryazanov, The Geometrical and Topological Theory of Functions and Mappings [in Russian], Naukova Dumka, Kiev, 2011.
V. Gutlyanskii, V. Ryazanov, U. Srebro, and E. Yakubov, “On recent advances in the Beltrami equations,” Ukr. Mat. Vesn., 7, No. 4, 467–515 (2010).
S. Hencl and P. Koskela, “Regularity of the inverse of a planar Sobolev homeomorphism,” Arch. Rat. Mech. Anal., 180, No. 1, 75–95 (2006).
S. L. Krushkal, Quasiconformal Mappings and Riemann Surfaces, Wiley, New York, 1979.
P.-J. Laurent, Approximation et Optimisation, Hermann, Paris, 1972.
O. Lehto and K. Virtanen, Quasiconformal Mappings in the Plane, Springer, New York, 1973.
T. V. Lomako, “To the theory of convergence and compactness for Beltrami equations,” Ukr. Mat. Zh., 63, No. 3, 341–349 (2011).
T. V. Lomako, “To the theory of convergence and compactness for Beltrami equations with constraints of the set-theoretic type,” Ukr. Mat. Zh., 63, No. 9, 1227–1240 (2011).
O. Martio, V. Ryazanov, U. Srebro, and E. Yakubov, Moduli in Modern Mapping Theory, Springer, New York, 2009.
D. Menchoff, “Sur les differentielles totales des fonctions univalentes,” Math. Ann., 105, 75–85 (1931).
S. P. Ponomarev, “N −1-property of mappings and the Luzin’s (N)-condition,” Mat. Zam., 58, 411–418 (1995).
W. Rudin, Function Theory in Polydiscs, Benjamin, New York, 1969.
V. I. Ryazanov, “Some questions of convergence and compactness for quasiconformal mappings,” in Theory of Mappings and Approximation of Functions [in Russian], edited by G. D. Suvorov, Naukova Dumka, Kiev, 1983, pp. 50–62.
V. I. Ryazanov, “Some questions of convergence and compactness for quasiconformal mappings,” Amer. Math. Soc. Transl., 131, No. 2, 7–19 (1986).
V. I. Ryazanov, Topological Aspects of the Theory of Quasiconformal Mappings [in Russian], Dissertation on Doctoral Degree (Phys.-Math. Sci.) IPMM NANU, Donetsk, 1993.
V. Ryazanov, U. Srebro, and E. Yakubov, “On strong solutions of the Beltrami equations,” Complex Var. Ellipt. Eq., 55, No. 1–3, 219–236 (2010).
S. Saks, Theory of the Integral, New York, Dover, 2005.
M. Schiffer and G. Schober, “A variational method for general families of quasiconformal mappings,” J. Anal. Math., 34, 240–264 (1978).
G. Schober, Univalent Functions, Lect. Notes Math., 478, Springer, Berlin, 1975.
A. Ukhlov, “Mappings generating the embeddings of Sobolev spaces,” Sib. Mat. Zh., 34, No. 1, 165–171 (1993).
S. K. Vodopyanov and A. Ukhlov, “Sobolev spaces and (P, Q)-quasiconformal mappings of Carnot groups,” Sib. Mat. Zh., 39, No. 4, 665–682 (1998).
Author information
Authors and Affiliations
Corresponding author
Additional information
Translated from Ukrains’kiĭ Matematychnyĭ Visnyk, Vol. 8, No. 4, pp. 513–536, October–November, 2011.
Rights and permissions
About this article
Cite this article
Gutlyanskiĭ, V.Y., Lomako, T.V. & Ryazanov, V.I. To the theory of variational method for Beltrami equations. J Math Sci 182, 37–54 (2012). https://doi.org/10.1007/s10958-012-0727-1
Received:
Published:
Issue Date:
DOI: https://doi.org/10.1007/s10958-012-0727-1