Semiglobal solvability of a class of planar vector fields of infinite type. (English) Zbl 0983.35036

Bergamasco, Adalberto P.; Meziani, Abdelhamid

Mat. Contemp. 18, 31-42 (2000).

The authors consider a special class of planar complex-valued vector fields \(L\) having the unit circle \(\Sigma\subset \mathbb{R}\times S^1\), \(\Sigma= \{0\}\times S^1\) as characteristic set. The vector field \(L\) is of infinite type along \(\Sigma\) and satisfies the famous Nirenberg-Treves condition for local solvability. Theorem 2.1 deals with a necessary and sufficient condition for analytic solvability of the equation \(Lu=f\) near \(\Sigma\), while in Theorem 3.1 it is shown that in general \(Lu= f\in C^\infty\) does not have \(C^\infty\) solutions in any neighbourhood of \(\Sigma\).

Reviewer: Petar Popivanov (Sofia)

MSC:

35F05	Linear first-order PDEs
35A05	General existence and uniqueness theorems (PDE) (MSC2000)

Keywords:

exponential Liouville number; Nirenberg-Treves condition; necessary and sufficient condition for analytic solvability