Solvability of a class of first order differential operators on the torus
The authors study global solvability and hypoellipticity on the $n$-torus of the equation $Pu=Lu+au+bu*=f$ where $L$ is a complex valued vector field and $u*$ is the complex conjugate of $u$. Because of the conjugation, $P$ is non-linear, and its study is new with respect to the most part of the preceding literature, see [\textit{A. P. Bergamasco} et al., J. Differ. Equations 264, No. 5, 3500--3526 (2018; Zbl 1395.35074)] for a recent survey.The authors extend the results already obtained for the equation in the 2-dimensional case by [\textit{A. P. Bergamasco} et al., J. Math. Anal. Appl. 416, No. 1, 166--180 (2014; Zbl 1302.35106)]. In particular, a complete study is given in the case when the coefficients of $P$ are constant, by including analysis of Gevrey solvability and hypoellipticity on the $n$-torus. The case of the variable coefficients is also considered. As standard for this kind of global problems, Diophantine conditions play an important role in the statements.
scientific article; zbMATH DE number 7369337
scientific article