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Some remarks on an existence problem for degenerate elliptic systems

Authors: Olli Martio, Vladimir Miklyukov and Matti Vuorinen
Journal: Proc. Amer. Math. Soc. 133 (2005), 1451-1458
MSC (2000): Primary 30C62; Secondary 35J70
Published electronically: November 22, 2004
MathSciNet review: 2111944
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Abstract: The system $au_x+ bu_y=v_y,\quad cu_x+du_y=-v_x ,$ which yields Beltrami's system if $b=c$, is considered. Under a condition for the coefficients $a,b,c,d $ a non-existence theorem is proved.

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  • 1. B. V. Boyarskiĭ, Generalized solutions of a system of differential equations of first order and of elliptic type with discontinuous coefficients, Mat. Sb. N.S. 43(85) (1957), 451–503 (Russian). MR 0106324
  • 2. Melkana A. Brakalova and James A. Jenkins, On solutions of the Beltrami equation, J. Anal. Math. 76 (1998), 67–92. MR 1676936,
  • 3. Ju. D. Burago and V. A. Zalgaller, \cyr Geometricheskie neravenstva, “Nauka” Leningrad. Otdel., Leningrad, 1980 (Russian). MR 602952
  • 4. E. A. Ščerbakov, Homeomorphic solutions of degenerate elliptic systems, Mathematical physics, No. 8 (Russian), Naukova Dumka, Kiev, 1970, pp. 187–190 (Russian). MR 0293239
  • 5. Guy David, Solutions de l’équation de Beltrami avec |𝜇|_{∞}=1, Ann. Acad. Sci. Fenn. Ser. A I Math. 13 (1988), no. 1, 25–70 (French). MR 975566,
  • 6. A. Džuraev, A system of Beltrami equations which are degenerate on a curve, Dokl. Akad. Nauk SSSR 185 (1969), 984–986 (Russian). MR 0241668
  • 7. F. R. Gantmacher, Matrizenrechnung. II. Spezielle Fragen und Anwendungen, Hochschulbücher für Mathematik, Bd. 37, VEB Deutscher Verlag der Wissenschaften, Berlin, 1959 (German). MR 0107647
    F. R. Gantmacher, Applications of the theory of matrices, Translated by J. L. Brenner, with the assistance of D. W. Bushaw and S. Evanusa, Interscience Publishers, Inc., New York; Interscience Publishers Ltd., London, 1959. MR 0107648
    F. R. Gantmacher, The theory of matrices. Vols. 1, 2, Translated by K. A. Hirsch, Chelsea Publishing Co., New York, 1959. MR 0107649
  • 8. V. GUTLYANSKII, O. MARTIO, T. SUGAWA AND M. VUORINEN: On the degenerate Beltrami equation, Reports of the Dept. of Math., Univ. of Helsinki, Preprint 282, 2001, 1-32, Trans. Amer. Math. Soc. (to appear).
  • 9. Juha Heinonen, Tero Kilpeläinen, and Olli Martio, Nonlinear potential theory of degenerate elliptic equations, Oxford Mathematical Monographs, The Clarendon Press, Oxford University Press, New York, 1993. Oxford Science Publications. MR 1207810
  • 10. O. MARTIO AND V.M. MIKLYUKOV: On existence and uniqueness of degenerate Beltrami equation, Reports of the Dept. of Math., Univ. of Helsinki, Preprint 347, 2003, 1-12.
  • 11. O. MARTIO, V.M. MIKLYUKOV AND M. VUORINEN: Functions monotone close to boundary, Reports of the Dept. of Math., Univ. of Helsinki, Preprint 330, March 2002, 1-20, Tohoku Math. J. (to appear).
  • 12. A. P. Mikhaĭlov, A mapping problem for a system of elliptic type, degenerate on the boundary, Sibirsk. Mat. Zh. 24 (1983), no. 3, 119–127 (Russian). MR 704163
  • 13. I. S. Ovčinnikov, The existence of mappings on the plane for degenerate first order elliptic systems, Dokl. Akad. Nauk SSSR 191 (1970), 526–529 (Russian). MR 0259355
  • 14. Uri Srebro and Eduard Yakubov, Branched folded maps and alternating Beltrami equations, J. Anal. Math. 70 (1996), 65–90. MR 1444258,
  • 15. William P. Ziemer, Weakly differentiable functions, Graduate Texts in Mathematics, vol. 120, Springer-Verlag, New York, 1989. Sobolev spaces and functions of bounded variation. MR 1014685

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Additional Information

Olli Martio
Affiliation: Department of Mathematics and Statistics, P.O. Box 68, FIN-00014, University of Helsinki, Finland

Vladimir Miklyukov
Affiliation: Department of Mathematics, Volgograd State University, 2 Prodolnaya, 30, Volgograd, 400062, Russia

Matti Vuorinen
Affiliation: Department of Mathematics, FIN-20014, University of Turku, Finland

Received by editor(s): June 1, 2003
Received by editor(s) in revised form: January 22, 2004
Published electronically: November 22, 2004
Communicated by: Richard A. Wentworth
Article copyright: © Copyright 2004 American Mathematical Society
The copyright for this article reverts to public domain 28 years after publication.