Elliptic planar vector fields with degeneracies

Meziani, Abdelhamid

doi:10.1090/S0002-9947-04-03658-X

Elliptic planar vector fields with degeneracies
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by Abdelhamid Meziani

Trans. Amer. Math. Soc. 357 (2005), 4225-4248

DOI: https://doi.org/10.1090/S0002-9947-04-03658-X

Published electronically: December 9, 2004

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Abstract:

This paper deals with the normalization of elliptic vector fields in the plane that degenerate along a simple and closed curve. The associated homogeneous equation $Lu=0$ is studied and an application to a degenerate Beltrami equation is given.

References

A. Bergamasco, P. Cordaro, and J. Hounie, Global properties of a class of vector fields in the plane, J. Differential Equations 74 (1988), no. 2, 179–199. MR 952894, DOI 10.1016/0022-0396(88)90001-0
A. Bergamasco, P. Cordaro, and G. Petronilho, Global Solvability for a class of complex vector fields on the two-torus, Preprint.
Adalberto P. Bergamasco and Abdelhamid Meziani, Semiglobal solvability of a class of planar vector fields of infinite type, Mat. Contemp. 18 (2000), 31–42 (English, with English and Portuguese summaries). VI Workshop on Partial Differential Equations, Part I (Rio de Janeiro, 1999). MR 1812861
A.Bergamasco and A. Meziani, Solvability near the characteristic set for a class of planar vector fields of infinite type, Preprint.
S. Berhanu and A. Meziani, On rotationally invariant vector fields in the plane, Manuscripta Math. 89 (1996), no. 3, 355–371. MR 1378599, DOI 10.1007/BF02567523
S. Berhanu and A. Meziani, Global properties of a class of planar vector fields of infinite type, Comm. Partial Differential Equations 22 (1997), no. 1-2, 99–142. MR 1434140, DOI 10.1080/03605309708821257
N. Bliev, Generalized analytic functions in fractional spaces, Pitman Monographs and Surveys in Pure and Applied Mathematics, vol. 86, Longman, Harlow, 1997. Translated from the 1985 Russian original by H. Begehr and R. Radok [Jens Rainer Maria Radok]. MR 1477905
P. Cordaro and X. Gong, Normalization of complex-valued planar vector fields which degenerate along a real curve, Preprint.
Abdelhamid Meziani, On real analytic planar vector fields near the characteristic set, Analysis, geometry, number theory: the mathematics of Leon Ehrenpreis (Philadelphia, PA, 1998) Contemp. Math., vol. 251, Amer. Math. Soc., Providence, RI, 2000, pp. 429–438. MR 1771284, DOI 10.1090/conm/251/03885
Abdelhamid Meziani, On planar elliptic structures with infinite type degeneracy, J. Funct. Anal. 179 (2001), no. 2, 333–373. MR 1809114, DOI 10.1006/jfan.2000.3695
L. Nirenberg and F. Treves, Solvability of a first order linear partial differential equation, Comm. Pure Appl. Math. 16 (1963), 331–351. MR 163045, DOI 10.1002/cpa.3160160308
François Trèves, Remarks about certain first-order linear PDE in two variables, Comm. Partial Differential Equations 5 (1980), no. 4, 381–425. MR 567779, DOI 10.1080/0360530800882143
François Trèves, Hypo-analytic structures, Princeton Mathematical Series, vol. 40, Princeton University Press, Princeton, NJ, 1992. Local theory. MR 1200459
I. N. Vekua, Generalized analytic functions, Pergamon Press, London-Paris-Frankfurt; Addison-Wesley Publishing Company, Inc., Reading, Mass., 1962. MR 0150320
W. L. Wendland, Elliptic systems in the plane, Monographs and Studies in Mathematics, vol. 3, Pitman (Advanced Publishing Program), Boston, Mass.-London, 1979. MR 518816