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), 14511458
MSC (2000):
Primary 30C62; Secondary 35J70
Published electronically:
November 22, 2004
MathSciNet review:
2111944
Fulltext PDF Free Access
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Abstract: The system which yields Beltrami's system if , is considered. Under a condition for the coefficients a nonexistence theorem is proved.
 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
(21 #5058)
 2.
Melkana
A. Brakalova and James
A. Jenkins, On solutions of the Beltrami equation, J. Anal.
Math. 76 (1998), 67–92. MR 1676936
(2000h:30029), http://dx.doi.org/10.1007/BF02786930
 3.
Ju.
D. Burago and V.
A. Zalgaller, Geometricheskie neravenstva, “Nauka”
Leningrad. Otdel., Leningrad, 1980 (Russian). MR 602952
(82d:52009)
 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
(45 #2316)
 5.
Guy
David, Solutions de l’équation de Beltrami avec
\Vert𝜇\Vert_{∞}=1, Ann. Acad. Sci. Fenn. Ser. A I Math.
13 (1988), no. 1, 25–70 (French). MR 975566
(90d:30058), http://dx.doi.org/10.5186/aasfm.1988.1303
 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
(39 #3007)
 7.
F.
R. Gantmacher, The theory of matrices. Vols. 1, 2, Translated
by K. A. Hirsch, Chelsea Publishing Co., 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, 132, 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
(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, 112.
 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, 120, 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
(84m:30075)
 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
(41 #3993)
 14.
Uri
Srebro and Eduard
Yakubov, Branched folded maps and alternating Beltrami
equations, J. Anal. Math. 70 (1996), 65–90. MR 1444258
(98f:30021), http://dx.doi.org/10.1007/BF02820441
 15.
William
P. Ziemer, Weakly differentiable functions, Graduate Texts in
Mathematics, vol. 120, SpringerVerlag, New York, 1989. Sobolev spaces
and functions of bounded variation. MR 1014685
(91e:46046)
 1.
 B.V. BOJARSKI: Generalized solutions of an elliptic system of first order with discontinuous coefficients (in Russian), Mat. Sb. 43, No. 4, 451503, 1957. MR 0106324 (21:5058)
 2.
 M.A. BRAKALOVA AND J.A. JENKINS: On solutions of the Beltrami equation. J. Anal. Math. 76 (1998), 6792. 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, 187190, 1970. MR 0293239 (45:2316)
 5.
 G. DAVID: Solutions de l'équation de Beltrami avec , Ann. Acad. Sci. Fenn. Ser. A I Math. 13, 2570, 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, 449452. (Russian) Dokl. Akad. Nauk SSSR 185 1969, 984986. 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, 132, 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, 112.
 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, 120, 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, 119127, 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, 526529, 1970. MR 0259355 (41:3993)
 14.
 U. SREBRO AND E. YAKUBOV: Branched folder maps and alternating Beltrami equations, J. Anal. Math. 70, 6590, 1996. MR 1444258 (98f:30021)
 15.
 W.P. ZIEMER: Weakly Differentiable Functions, Graduate Texts in Mathematics 120, SpringerVerlag, 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, FIN00014, 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, FIN20014, University of Turku, Finland
Email:
vuorinen@csc.fi
DOI:
http://dx.doi.org/10.1090/S0002993904076956
PII:
S 00029939(04)076956
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.
