Extremal problem for functional of area

T.V. Lomako (Institute of Applied Mathematics and Mechanics, NAS of Ukraine)
R.R. Salimov (Institute of Applied Mathematics and Mechanics, NAS of Ukraine)

Abstract


In the present article, the generalization of Lavrent'ev problem about the estimate of area of image of the circle is studied. It is shown that the above estimate in the investigated class is precise.

References


Lavrentiev M.A. Variational method in boundary problems for sets of equations of elliptic type, 1962; (in Russian)

Martio O., Ryazanov V., Srebro U., Yakubov E. "Mappings with finite length distortion", J. d'Anal. Math., 2004; 93: pp. 215-236. doi:10.1007/BF02789308

Martio O., Ryazanov V., Srebro U., Yakubov E. "On Q-homeomorphisms", Ann. Acad. Sci. Fenn., 2005; 30: pp. 49-69.

Martio O., Ryazanov V., Srebro U., Yakubov E. "Q-homeomorphisms", Contemp. Math., 2006; 364: pp. 193-203. doi:10.1090/conm/364/06685

Vaisala J. Lectures on n-Dimensional Quasiconformal Mappings, LNM 229, Springer-Verlag, 1971.

Ryazanov V., Srebro U., Yakubov E. "On ring solutions of Beltrami equations", J. d'Anal. Math., 2005; 96: pp. 117-150. doi:10.1007/BF02787826

Martio O., Rickman S., Vaisala J. "Definitions for quasiregular mappings", Ann. Acad. Sci. Fenn. Ser. A1. Math., 1969.

Hesse J. "A $$$p$$$-extremal length and $$$p$$$-capacity equality", Ark. Math., 1975; 13: pp. 131-144. doi:10.1007/BF02386202

Gehring F. "Quasiconformal mappings", Complex Analysis and its Applications. Vol. 2, Int. Atomic Energy Agency, 1976.

Shlyk V.A. "On the equality between $$$p$$$-capacity and $$$p$$$-modulus", Siberian Math. J., 1993; 34(6): pp. 1196-1200. doi:10.1007/BF00973485

Bojarski B., Gutlyanskii V., Martio O., Ryazanov V. Infinitesimal Geometry of Quasiconformal and Bi-Lipschitz Mappings in the Plane, EMS, 2013. doi:10.4171/122

Ryazanov V., Sevostianov E. "Equicontinuous classes of ring Q-homeomorphisms", Siberian Math. J., 2007; 48(6): pp. 1361-1375. (in Russian) doi:10.1007/s11202-007-0111-4




DOI: https://doi.org/10.15421/240912

  

Refbacks

  • There are currently no refbacks.


Copyright (c) 2009 T.V. Lomako, R.R. Salimov

Creative Commons License
This work is licensed under a Creative Commons Attribution 4.0 International License.


Registered in


ISSN (Online): 2664-5009
ISSN (Print): 2664-4991
DNU