×

zbMATH — the first resource for mathematics

Gevrey solvability near the characteristic set for a class of planar complex vector fields of infinite type. (English) Zbl 1173.35300
The authors consider the complex vector field
\[ L= \partial/\partial t+ (a(x)+ ib(x))\partial/\partial x \] defined in \(\Omega= (-\lambda,\lambda)\times S^1\), where \(\lambda> 0\), and \(S^1\) is the unit circle in the variable \(t\). Under the assumption \[ (a+ ib)(x)= x^n a_0(x)+ ix^m b_0(x),\quad n,m\geq 1\quad (a_0+ ib_0)(0)\neq 0, \] with \(a_0(x)\) and \(b_0(x)\) in the Gevrey class \(G^s(-\lambda, \lambda)\), \(s\geq 1\), the authors study the solvability of the equation \(Lu= f\) in Gevrey classes on \(\Omega\). Detailed results are given, and several examples/counterexamples are provided, extending A. P. Bergamasco and A. Meziani [Ann. Inst. Fourier 55, No. 1, 77–112 (2005; Zbl 1063.35051)].

MSC:
35A05 General existence and uniqueness theorems (PDE) (MSC2000)
58J05 Elliptic equations on manifolds, general theory
PDF BibTeX XML Cite
Full Text: DOI
References:
[1] Bergamasco, A.P., Remarks about global analytic hypoellipticity, Trans. amer. math. soc., 351, 4113-4126, (1999) · Zbl 0932.35046
[2] Bergamasco, A.P.; Cordaro, P.D.; Hounie, J., Global properties of a class of vector fields in the plane, J. differential equations, 74, 179-199, (1988) · Zbl 0662.58021
[3] Bergamasco, A.P.; Cordaro, P.D.; Malagutti, P.L.A., Globally hypoelliptic systems of vector fields, J. funct. anal., 114, 267-285, (1993) · Zbl 0777.58041
[4] Bergamasco, A.P.; Cordaro, P.D.; Petronilho, G., Global solvability for a class of complex vector fields on the two-torus, Comm. partial differential equations, 29, 785-819, (2004) · Zbl 1065.35088
[5] Bergamasco, A.P.; Dattori da Silva, P.L., Global solvability for a special class of vector fields on the torus, (), 11-20 · Zbl 1108.35026
[6] Bergamasco, A.P.; Dattori da Silva, P.L., Solvability in the large for a class of vector fields on the torus, J. math. pures appl., 86, 427-447, (2006) · Zbl 1157.35304
[7] Bergamasco, A.P.; Kirilov, A., Global solvability for a class of overdetermined systems, J. funct. anal., 252, 603-629, (2007) · Zbl 1158.58011
[8] Bergamasco, A.P.; Meziani, A., Solvability near the characteristic set for a class of planar vector fields of infinite type, Ann. inst. Fourier (Grenoble), 55, 77-112, (2005) · Zbl 1063.35051
[9] Bergamasco, A.P.; Nunes, W.V.L.; Zani, S.L., Global analytic hypoellipticity and pseudoperiodic functions, Mat. contemp., 18, 43-57, (2000) · Zbl 0979.35036
[10] Bergamasco, A.P.; Nunes, W.V.L.; Zani, S.L., Global properties of a class of overdetermined systems, J. funct. anal., 200, 31-63, (2003)
[11] Bergamasco, A.P.; Petronilho, G., Closedness of the range for vector fields on the torus, J. differential equations, 154, 132-139, (1999) · Zbl 0926.35030
[12] Bergamasco, A.P.; Zani, S.L., Prescribing analytic singularities for solutions of a class of vector fields on the torus, Trans. amer. math. soc., 357, 4159-4174, (2005) · Zbl 1077.35004
[13] Bergamasco, A.P.; Zani, S.L., Globally analytic hypoelliptic vector fields on compact surfaces, Proc. amer. math. soc., 136, 1305-1310, (2008) · Zbl 1139.35038
[14] Bergamasco, A.P.; Zani, S.L., Global analytic regularity for structures of co-rank one, Comm. partial differential equations, 33, 933-941, (2008) · Zbl 1153.35006
[15] Berhanu, S.; Cordaro, P.D.; Hounie, J., An introduction to involutive structures, (2008), Cambridge Univ. Press · Zbl 1151.35011
[16] Berhanu, S.; Meziani, A., Global properties of a class of planar vector fields of infinite type, Comm. partial differential equations, 22, 99-142, (1997) · Zbl 0882.35029
[17] P.A.S. Caetano, P.D. Cordaro, Gevrey solvability and Gevrey regularity in differential complexes associated to locally integrable structures, Trans. Amer. Math. Soc., in press · Zbl 1217.35042
[18] Dattori da Silva, P.L., Nonexistence of global solutions for a class of complex vector fields on two-torus, J. math. anal. appl., 351, 543-555, (2009) · Zbl 1173.35406
[19] Gramchev, T.; Popivanov, P.; Yoshino, M., Global properties in spaces of generalized functions on the torus for second order differential operators with variable coefficients, Rend. sem. mat. univ. politec. Torino, 51, 2, 146-172, (1993) · Zbl 0824.35027
[20] Hörmander, L., The analysis of linear partial differential operators I, (1984), Springer-Verlag
[21] Hörmander, L., The analysis of linear partial differential operators IV, (1984), Springer-Verlag
[22] Komatsu, H., Ultradistributions. II. the kernel theorem and ultradistributions with support in a submanifold, J. fac. sci. univ. Tokyo sect. IA math., 24, 3, 607-628, (1977) · Zbl 0385.46027
[23] Lascar, B.; Lerner, N., Propagation des singularités Gevrey pour l’équation de cauchy – riemann dégénérée, Israel J. math., 124, 299-312, (2001) · Zbl 1017.35123
[24] Lascar, B.; Lerner, N., Propagation des singularités Gevrey pour des équations de type principal satisfaisant la condition (P), Osaka J. math., 39, 511-521, (2002) · Zbl 1028.35007
[25] Meziani, A., Elliptic planar vector fields with degeneracies, Trans. amer. math. soc., 357, 4225-4248, (2005) · Zbl 1246.35090
[26] Nirenberg, L.; Treves, F., Solvability of a first order linear partial differential equations, Comm. pure appl. math., 16, 331-351, (1963) · Zbl 0117.06104
[27] Petzsche, H.-J., On E. Borel’s theorem, Math. ann., 282, 299-313, (1988) · Zbl 0633.46033
[28] Rodino, L., Linear partial differential operators in Gevrey spaces, (1993), World Scientific · Zbl 0869.35005
[29] Schmets, J.; Valdivia, M., Extension maps in ultradifferentiable and ultraholomorphic function spaces, Studia math., 143, 3, 221-250, (2000) · Zbl 0972.46013
[30] Treves, F., Hypo-analytic structures. local theory, (1992), Princeton Univ. Press Princeton, NJ · Zbl 0787.35003
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.