×

zbMATH — the first resource for mathematics

Normalization of complex-valued planar vector fields which degenerate along a real curve. (English) Zbl 1129.35419
Summary: Taking as a start point the recent article of A. Meziani [J. Funct. Anal. 179, No. 2, 333–373 (2001; Zbl 0973.35083)], we present several results concerning the normalization of a class of complex vector fields in the plane which degenerate along a real curve. We mainly deal with operators with finite regularity and analyze both the local situation as well as the case of normalization near a circle. Some related questions (e.g., on semi-global solvability and on the normalization of a class of generalized Mizohata operators) are also discussed.

MSC:
35J70 Degenerate elliptic equations
35A20 Analyticity in context of PDEs
37G05 Normal forms for dynamical systems
PDF BibTeX XML Cite
Full Text: DOI
References:
[1] Ahern, P.; Rosay, J.-P., Entire functions, in the classification of differentiable germs tangent to the identity, in one or two variables, Trans. amer. math. soc., 347, 2, 543-572, (1995) · Zbl 0815.30018
[2] A.P. Bergamasco, A. Meziani, Semiglobal solvability of a class of planar vector fields of infinite type, VI Workshop on Partial Differential Equations, Part I, Rio de Janeiro, 1999; Mat. Contemp. Vol. 18 (2000) pp. 31-42. · Zbl 0983.35036
[3] Brjuno, A.D., Analytic form of differential equations, Trans. Moscow math. soc., 25, 131-288 (1973), (1971)
[4] L. Hörmander, The analysis of linear partial differential operators, I and IV. Grundlehren der Mathematischen Wissenschaften (Fundamental Principles of Mathematical Sciences), Vol. 256, 275. Springer, Berlin, 1983, 1985.
[5] Hounie, J.G., Local solvability of first order linear operators with Lipschitz coefficients, Duke math. J., 62, 467-477, (1991) · Zbl 0731.35025
[6] D. Kim, Ph.D. Thesis, Rutgers University, 1981.
[7] Meziani, A., On planar elliptic structures with infinite type degeneracy, J. funct. anal., 179, 2, 333-373, (2001) · Zbl 0973.35083
[8] Ninomiya, H., A necessary and sufficient condition of local integrability, J. math. Kyoto univ., 39, 4, 685-696, (1999) · Zbl 0960.35017
[9] L. Nirenberg, Lectures on Linear Partial Differential Equations, Reg. Conference Series in Mathematics, Vol. 17, American Mathematical Society, Providence, RI 1973. · Zbl 0267.35001
[10] Pérez Marco, R.; Yoccoz, J.-C., Germes de feuilletages holomorphes à holonomie prescrite, Complex analytic methods in dynamical systems, Rio de Janeiro, 1992; astérisque, 222, 7, 345-371, (1994) · Zbl 0809.32008
[11] Siegel, C.L., Iteration of analytic functions, Ann. of math., 43, 2, 607-612, (1942) · Zbl 0061.14904
[12] Sjöstrand, J., Note on a paper of F. treves concerning mizohata type operators, Duke math. J., 47, 3, 601-608, (1980) · Zbl 0471.35076
[13] Treves, F., Hypoelliptic partial differential equations of principal type. sufficient conditions and necessary conditions, Comm. pure appl. math., 24, 631-670, (1971) · Zbl 0234.35019
[14] Treves, F., Remarks about certain first-order linear PDE in two variables, Comm. partial differential equations, 5, 4, 381-425, (1980) · Zbl 0523.35012
[15] Treves, F., Hypo-analytic structures (local theory), Princeton mathematical series, Vol. 40, (1992), Princeton University Press Princeton, NJ · Zbl 0787.35003
[16] Vekua, I.N., Generalized analytic functions, (1962), Pergamon Press, London-Paris-Frankfurt, Addison-Wesley Publishing Co., Inc Reading, MA · Zbl 0127.03505
[17] Voronin, V.M., Analytic classification of germs of conformal mappings \((C,0)→ (C,0)\), Funct. anal. appl., 15, 1, 1-13, (1981) · Zbl 0463.30010
[18] Yoccoz, J.-C., Théorème de Siegel, nombres de bruno et polynômes quadratiques, Petits diviseurs en dimension 1, astérisque, 231, 3-88, (1995)
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. It attempts to reflect the references listed in the original paper as accurately as possible without claiming the completeness or perfect precision of the matching.