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

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