Global hypoellipticity of a class of second order operators. (English) Zbl 0811.35016

Bergamasco, Adalberto P.; Zani, Sérgio Luís

Can. Math. Bull. 37, No. 3, 301-305 (1994).

Summary: We show that almost all perturbations \(P - \lambda\), \(\lambda \in \mathbb{C}\), of an arbitrary constant coefficient partial differential operator \(P\) are globally hypoelliptic on the torus. We also give a characterization of the values \(\lambda \in \mathbb{C}\) for which the operator \(D^ 2_ t - 2D^ 2_ x - \lambda\) is globally hypoelliptic; in particular, we show that the addition of a term of order zero may destroy the property of global hypoellipticity of operators of principal type, contrary to that happens with the usual (local) hypoellipticity.

MSC:

35H10	Hypoelliptic equations
35B10	Periodic solutions to PDEs
35B65	Smoothness and regularity of solutions to PDEs

Keywords:

local hypoellipticity; Liouville number; global hypoellipticity