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A Note about Analytic Solvability of Complex Planar Vector Fields with Degeneracies

Published online by Cambridge University Press:  20 November 2018

Paulo L. Dattori da Silva
Paulo L. Dattori da Silva*
Affiliation:
Universidade de São Paulo, Instituto de Ciências Matemáticas e de Computação, Departamento de Matemática, Caixa Postal 668, São Carlos - SP, 13560-970 Brazile-mail: [email protected]
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Abstract

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This paper deals with the analytic solvability of a special class of complex vector fields defined on the real plane, where they are tangent to a closed real curve, while off the real curve, they are elliptic.

Keywords

35A0158Jxxsemi-global solvabilityanalytic solvabilitynormalizationcomplex vector fieldscondition (P)
Type
Research Article
Information
Canadian Mathematical Bulletin , Volume 54 , Issue 2 , 01 June 2011 , pp. 249 - 254
Copyright
Copyright © Canadian Mathematical Society 2011

References

[1] Bergamasco, A. P., Cordaro, P. D., and Petronilho, G., Global solvability for a class of complex vector fields on the two-torus. Comm. Partial Differential Equations 29(2004), no. 5–6, 785–819. doi:10.1081/PDE-120037332Google Scholar
[2] Bergamasco, A. P. and Dattori da Silva, P. L., Global solvability for a special class of vector fields on the torus. In: Recent progress on some problems in several complex variables and partial differential equations, Contemp. Math., 400, American Mathematical Society, Providence, RI, 2006, pp. 11–20.Google Scholar
[3] Bergamasco, A. P. and Dattori da Silva, P. L., Solvability in the large for a class of vector fields on the torus. J. Math. Pures Appl. 86(2006), no. 5, 427–447.Google Scholar
[4] Bergamasco, A. P., Dattori da Silva, P. L., and Ebert, M. R., Gevrey solvability near the characteristic set for a class of planar complex vector fields of infinite type. J. Differential Equations 246(2009), no. 4, 1673–1702. doi:10.1016/j.jde.2008.10.028Google Scholar
[5] Bergamasco, A. P. and Meziani, A., Semiglobal solvability of a class of planar vector fields of infinite type. Mat. Contemp. 18(2000), 31–42.Google Scholar
[6] Bergamasco, A. P. and Meziani, A., Solvability near the characteristic set for a class of planar vector fields of infinite type. Ann. Inst. Fourier (Grenoble) 55(2005), no. 1, 77–112.Google Scholar
[7] Bergamasco, A. P. and Petronilho, G., Closedness of the range for vector fields on the torus. J. Differential Equations 154(1999), no. 1, 132–139. doi:10.1006/jdeq.1998.3563Google Scholar
[8] Berhanu, S. and Meziani, A., Global properties of a class of planar vector fields of infinite type. Comm. Partial Differential Equations 22(1997), no. 1–2, 99–142. doi:10.1080/03605309708821288Google Scholar
[9] Cerniauskas, W. A. and Kirilov, A., Ck solvability near the characteristic set for a class of vector fields of infinite type. Mat. Contemp. 36(2009), 91–106.Google Scholar
[10] Cordaro, P. D. and Gong, X., Normalization of complex-valued planar vector fields which degenerate along a real curve. Adv. Math. 184(2004), no. 1, 89–118. doi:10.1016/S0001-8708(03)00139-7Google Scholar
[11] Dattori da Silva, P. L., Nonexistence of global solutions for a class of complex vector fields on two-torus. J. Math. Anal. Appl. 351(2009), no. 2, 543–555. doi:10.1016/j.jmaa.2008.10.039Google Scholar
[12] Dattori da Silva, P. L., Ck-Solvability near the characteristic set for a class of planar complex vector fields of infinite type. Ann. Mat. Pura Appl. 189(2010), no. 3, 403–413. doi:10.1007/s10231-009-0115-8Google Scholar
[13] Hörmander, L., Pseudodifferential operators of principal type. In: Singularities in boundary value problems (Proc. NATO Adv. Study Inst., Maratea, 1980), NATO Adv. Study Inst. Ser. C: Math. Phys. Sci., 65, Reidel, Dordrecht-Boston, MA, 1981, pp. 69–96.Google Scholar
[14] Meziani, A., On planar elliptic structures with infinite type degeneracy. J. Funct. Anal. 179(2001), no. 2, 333–373. doi:10.1006/jfan.2000.3695Google Scholar
[15] Meziani, A., Elliptic planar vector fields with degeneracies. Trans. Amer. Math. Soc. 357(2005), no. 10, 4225–4248. doi:10.1090/S0002-9947-04-03658-XGoogle Scholar
[16] Nirenberg, L. and Treves, F., Solvability of a first order linear partial differential equation. Comm. Pure Appl. Math. 16(1963), 331–351. doi:10.1002/cpa.3160160308Google Scholar