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Some remarks on an existence problem for degenerate elliptic systems

Author(s): Olli Martio; Vladimir Miklyukov; Matti Vuorinen
Journal: Proc. Amer. Math. Soc. 133 (2005), 1451-1458.
MSC (2000): Primary 30C62; Secondary 35J70
Posted: November 22, 2004
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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.


References:

1.
B.V. BOJARSKI: Generalized solutions of an elliptic system of first order with discontinuous coefficients (in Russian), Mat. Sb. 43, No. 4, 451-503, 1957. MR 0106324 (21:5058)
2.
M.A. BRAKALOVA AND J.A. JENKINS: On solutions of the Beltrami equation. J. Anal. Math. 76 (1998), 67-92. MR 1676936 (2000h:30029)

3.
YU. BURAGO AND V. A. ZALGALLER: Geometric inequalities, Leningrad, "Nauka", 1980 (in Russian). MR 0602952 (82d:52009)
4.
E.A. CHTCHERBAKOV (SCERBAKOV): Homeomorphic solutions of degenerate elliptic systems (in Russian), Math. Phys., Kiev, Naukova Dumka 8, 187-190, 1970. MR 0293239 (45:2316)

5.
G. DAVID: Solutions de l'équation de Beltrami avec $\Vert\mu\Vert _{\infty}=1$, Ann. Acad. Sci. Fenn. Ser. A I Math. 13, 25-70, 1988. MR 0975566 (90d:30058)
6.
A. DZURAEV: On a Beltrami system of equations degenerating on a line, - Soviet Math. Dokl. 10, No. 2, 1969, 449-452. (Russian) Dokl. Akad. Nauk SSSR 185 1969, 984-986. MR 0241668 (39:3007)

7.
F.R. GANTMACHER: Theory of Matrices, Chelsea, New York, 1959. MR 0107649 (21:6372c)
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.
J. HEINONEN, T. KILPELÄINEN AND O. MARTIO: Nonlinear Potential Theory of Degenerate Elliptic Equations, Oxford Mathematical Monographs, Oxford University Press, Oxford, 1993. MR 1207810 (94e:31003)
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. MIHAILOV: A problem for a mapping by solutions of elliptic system degenerating on boundary (in Russian), Sibirsk. Mat. Zh. 24, No. 3, 119-127, 1983. MR 0704163 (84m:30075)
13.
I.S. OVCHINNIKOV: On existence of plane mappings for degenerate elliptic systems of the first order (in Russian), Dokl. AN SSSR 191, No. 3, 526-529, 1970. MR 0259355 (41:3993)
14.
U. SREBRO AND E. YAKUBOV: Branched folder maps and alternating Beltrami equations, J. Anal. Math. 70, 65-90, 1996. MR 1444258 (98f:30021)
15.
W.P. ZIEMER: Weakly Differentiable Functions, Graduate Texts in Mathematics 120, Springer-Verlag, New York, 1989. MR 1014685 (91e:46046)

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Additional Information:

Olli Martio
Affiliation: Department of Mathematics and Statistics, P.O. Box 68, FIN-00014, University of Helsinki, Finland
Email: martio@cc.helsinki.fi

Vladimir Miklyukov
Affiliation: Department of Mathematics, Volgograd State University, 2 Prodolnaya, 30, Volgograd, 400062, Russia
Email: miklyuk@mail.ru

Matti Vuorinen
Affiliation: Department of Mathematics, FIN-20014, University of Turku, Finland
Email: vuorinen@csc.fi

DOI: 10.1090/S0002-9939-04-07695-6
PII: S 0002-9939(04)07695-6
Received by editor(s): June 1, 2003
Received by editor(s) in revised form: January 22, 2004
Posted: November 22, 2004
Communicated by: Richard A. Wentworth
Copyright of article: Copyright 2004, American Mathematical Society
The copyright for this article reverts to public domain after 28 years from publication.


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