On the degenerate Beltrami equation

Gutlyanskiĭ, V.; Martio, O.; Sugawa, T.; Vuorinen, M.

doi:10.1090/S0002-9947-04-03708-0

On the degenerate Beltrami equation
HTML articles powered by AMS MathViewer

by V. Gutlyanskiĭ, O. Martio, T. Sugawa and M. Vuorinen

Trans. Amer. Math. Soc. 357 (2005), 875-900

DOI: https://doi.org/10.1090/S0002-9947-04-03708-0

Published electronically: October 19, 2004

PDF | Request permission

Abstract:

We study the well-known Beltrami equation under the assumption that its measurable complex-valued coefficient $\mu (z)$ has the norm $\|\mu \|_\infty =1.$ Sufficient conditions for the existence of a homeomorphic solution to the Beltrami equation on the Riemann sphere are given in terms of the directional dilatation coefficients of $\mu .$ A uniqueness theorem is also proved when the singular set $\operatorname {Sing} (\mu )$ of $\mu$ is contained in a totally disconnected compact set with an additional thinness condition on $\operatorname {Sing}(\mu ).$

References

Lars V. Ahlfors, Lectures on quasiconformal mappings, Van Nostrand Mathematical Studies, No. 10, D. Van Nostrand Co., Inc., Toronto, Ont.-New York-London, 1966. Manuscript prepared with the assistance of Clifford J. Earle, Jr. MR 0200442
Cabiria Andreian Cazacu, Influence of the orientation of the characteristic ellipses on the properties of the quasiconformal mappings, Proceedings of the Romanian-Finnish Seminar on Teichmüller Spaces and Quasiconformal Mappings (Braşov, 1969) Publ. House of the Acad. of the Socialist Republic of Romania, Bucharest, 1971, pp. 65–85. MR 0318478
Kari Astala, Tadeusz Iwaniec, Pekka Koskela, and Gaven Martin, Mappings of BMO-bounded distortion, Math. Ann. 317 (2000), no. 4, 703–726. MR 1777116, DOI 10.1007/PL00004420
P. P. Belinskiĭ, Obshchie svoĭ stva kvazikonformnykh otobrazheniĭ, Izdat. “Nauka” Sibirsk. Otdel., Novosibirsk, 1974 (Russian). MR 0407275
Lipman Bers, Uniformization by Beltrami equations, Comm. Pure Appl. Math. 14 (1961), 215–228. MR 132175, DOI 10.1002/cpa.3160140304
Melkana A. Brakalova and James A. Jenkins, On solutions of the Beltrami equation, J. Anal. Math. 76 (1998), 67–92. MR 1676936, DOI 10.1007/BF02786930
Zhiguo Chen, Estimates on $\mu (z)$-homeomorphisms of the unit disk, Israel J. Math. 122 (2001), 347–358. MR 1826507, DOI 10.1007/BF02809907
Guy David, Solutions de l’équation de Beltrami avec $\|\mu \|_\infty =1$, Ann. Acad. Sci. Fenn. Ser. A I Math. 13 (1988), no. 1, 25–70 (French). MR 975566, DOI 10.5186/aasfm.1988.1303
F. W. Gehring, A remark on the moduli of rings, Comment. Math. Helv. 36 (1961), 42–46. MR 140682, DOI 10.1007/BF02566891
Yasuhiro Gotoh and Masahiko Taniguchi, A condition of quasiconformal extendability, Proc. Japan Acad. Ser. A Math. Sci. 75 (1999), no. 4, 58–60. MR 1701526
David A. Herron, Xiang Yang Liu, and David Minda, Ring domains with separating circles or separating annuli, J. Analyse Math. 53 (1989), 233–252. MR 1014988, DOI 10.1007/BF02793416
T. Iwaniec and G. Martin, The Beltrami equation – In memory of Eugenio Beltrami (1835–1900), 100 years on, Memoires of the AMS, American Mathematical Society, to appear.
V. Ī. Kruglikov, The existence and uniqueness of mappings that are quasiconformal in the mean, Metric questions of the theory of functions and mappings, No. IV (Russian), Izdat. “Naukova Dumka”, Kiev, 1973, pp. 123–147 (Russian). MR 0344462
Olli Lehto, Homeomorphisms with a given dilatation, Proceedings of the Fifteenth Scandinavian Congress (Oslo, 1968) Lecture Notes in Mathematics, Vol. 118, Springer, Berlin, 1970, pp. 58–73. MR 0260997
Olli Lehto, Remarks on generalized Beltrami equations and conformal mappings, Proceedings of the Romanian-Finnish Seminar on Teichmüller Spaces and Quasiconformal Mappings (Braşov, 1969) Publ. House of the Acad. of the Socialist Republic of Romania, Bucharest, 1971, pp. 203–214. MR 0306489
O. Lehto and K. I. Virtanen, Quasiconformal mappings in the plane, 2nd ed., Die Grundlehren der mathematischen Wissenschaften, Band 126, Springer-Verlag, New York-Heidelberg, 1973. Translated from the German by K. W. Lucas. MR 0344463, DOI 10.1007/978-3-642-65513-5
Vladimir G. Maz’ja, Sobolev spaces, Springer Series in Soviet Mathematics, Springer-Verlag, Berlin, 1985. Translated from the Russian by T. O. Shaposhnikova. MR 817985, DOI 10.1007/978-3-662-09922-3
V. M. Mīkljukov and G. D. Suvorov, The existence and uniqueness of quasiconformal mappings with unbounded characteristics, Studies in the theory of functions of a complex variable and its applications (Russian and Ukrainian), Vidannja Īnst. Mat. Akad. Nauk Ukraïn. RSR, Kiev, 1972, pp. 45–53 (Russian). MR 0338361
I. N. Pesin, Mappings which are quasiconformal in the mean, Dokl. Akad. Nauk SSSR 187 (1969), 740–742 (Russian). MR 0249613
Edgar Reich and Hubert R. Walczak, On the behavior of quasiconformal mappings at a point, Trans. Amer. Math. Soc. 117 (1965), 338–351. MR 176070, DOI 10.1090/S0002-9947-1965-0176070-9
V. Ryazanov, U. Srebro, and E. Yakubov, BMO-quasiconformal mappings, J. Anal. Math. 83 (2001), 1–20. MR 1828484, DOI 10.1007/BF02790254
Uri Srebro and Eduard Yakubov, $\mu$-homeomorphisms, Lipa’s legacy (New York, 1995) Contemp. Math., vol. 211, Amer. Math. Soc., Providence, RI, 1997, pp. 473–479. MR 1477004, DOI 10.1090/conm/211/02837
G. D. Suvorov, Semeĭ stva ploskikh topologicheskikh otobrazheniĭ, Izdat. Sibirsk. Otdel. Akad. Nauk SSSR, Novosibirsk, 1965 (Russian). MR 0199425
Pekka Tukia, Compactness properties of $\mu$-homeomorphisms, Ann. Acad. Sci. Fenn. Ser. A I Math. 16 (1991), no. 1, 47–69. MR 1127696, DOI 10.5186/aasfm.1991.1628