Abstract
We consider the solvability problem for the equation \(f_{\bar z} \) = v(z, f(z))f z , where the function v(z,w) of two variables may be close to unity. Such equations are called quasilinear Beltrami-type equations with ellipticity degeneration. We prove that, under some rather general conditions on v(z,w), the above equation has a regular homeomorphic solution in the Sobolev classW 1,1loc . Moreover, such solutions f satisfy the inclusion f −1 ∈ W 1,2loc .
This is a preview of subscription content, log in via an institution to check access.
Similar content being viewed by others
References
B. V. Boyarskii, “Generalized solutions of a system of first-order differential equations of elliptic type with discontinuous coefficients,” Mat. Sb. 43(4), 451–503 (1957).
V. G. Maz’ya, Sobolev Spaces (Spannuluser-Verlag, Berlin, 1985).
O. Lehto and K. Virtanen, Quasiconformal Mappings in the Plane, in Grundlehren Math. Wiss. (Spannuluser, New York, 1973), Vol. 126.
M. A. Brakalova and J. A. Jenkins, “On solutions of the Beltrami equation,” J. Anal. Math. 76(1), 67–92 (1998).
V. Gutlyanskii, O. Martio, T. Sugawa, and M. Vuorinen, “On the degenerate Beltrami equation,” Trans. Amer. Math. Soc. 357(3), 875–900 (2005).
O. Martio, V. Ryazanov, U. Srebro, and E. Yakubov, Moduli in Modern Mapping Theory, in Spannuluser Monogr. Math. (Spannuluser, New York, 2009).
O. Martio and V. Miklyukov, “On existence and uniqueness of degenerate Beltrami equation,” Complex Var. Theory Appl. 49(7–9), 647–656 (2004).
V. Ryazanov, U. Srebro, and E. Yakubov, “On convergence theory for Beltrami equations,” Ukr. Mat. Visn. 5(4), 524–535 (2008).
F. John and L. Nirenberg, “On functions of bounded mean oscillation,” Comm. Pure Appl. Math. 14(3), 415–426 (1961).
C. J. Bishop, V. Ya. Gutlyanskii, O. Martio, and M. Vuorinen, “On conformal dilatation in space,” Int. J. Math. Math. Sci. 2003(22), 1397–1420 (2003).
L. Ahlfors, Lectures on Quasiconformal Mappings (Princeton, NJ, 1966; Mir, Moscow, 1969).
Yu. G. Reshetnyak, Spatial Mappings with Bounded Distorsion (Nauka, Novosibirsk, 1982) [in Russian].
S. P. Ponomarev, “The N −1-property of maps and Luzin’s condition (N),” Mat. Zametki 58(3), 411–418 (1995) [Math. Notes 58 (3–4), 960–965 (1995)].
B. V. Shabat, “On the theory of quasiconformal mappings in space,” Dokl. Akad. Nauk SSSR 132(5), 1045–1048 (1960) [Soviet Math. Dokl. 1, 730–733 (1960)].
V. A. Zorich, “Admissible order of growth of the quasiconformality characteristic in Lavrentév’s theorem,” Dokl. Akad. Nauk SSSR 181(3), 530–533 (1968) [SovietMath. Dokl. 9, 866–869 (1968)].
Author information
Authors and Affiliations
Corresponding author
Additional information
Original Russian Text © E. A. Sevost’yanov, 2011, published in Matematicheskie Zametki, 2011, Vol. 90, No. 3, pp. 445–453.
Rights and permissions
About this article
Cite this article
Sevost’yanov, E.A. On quasilinear Beltrami-type equations with degeneration. Math Notes 90, 431 (2011). https://doi.org/10.1134/S0001434611090112
Received:
Accepted:
Published:
DOI: https://doi.org/10.1134/S0001434611090112