an:05529545
Zbl 1173.35300
Bergamasco, Adalberto P.; da Silva, Paulo L. Dattori; Ebert, Marcelo R.
Gevrey solvability near the characteristic set for a class of planar complex vector fields of infinite type
EN
J. Differ. Equations 246, No. 4, 1673-1702 (2009).
00247037
2009
j
35A05 58J05
Gevrey classes; complex vector fields; global solvability; semi-global solvability; Fourier series; Whitney extension
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 \textit{A. P. Bergamasco} and \textit{A. Meziani} [Ann. Inst. Fourier 55, No.~1, 77--112 (2005; Zbl 1063.35051)].
Luigi Rodino (Torino)
Zbl 1063.35051