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International Journal of Differential Equations
Volume 2013 (2013), Article ID 526390, 8 pages
http://dx.doi.org/10.1155/2013/526390
Research Article

Some Properties of Solutions to Weakly Hypoelliptic Equations

Universität Potsdam, Institut für Mathematik, Am Neuen Palais 10, 14469 Potsdam, Germany

Received 21 May 2013; Accepted 4 July 2013

Academic Editor: Qi Zhang

Copyright © 2013 Christian Bär. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.

Abstract

A linear different operator is called weakly hypoelliptic if any local solution of is smooth. We allow for systems, that is, the coefficients may be matrices, not necessarily of square size. This is a huge class of important operators which coverall elliptic, overdetermined elliptic, subelliptic, and parabolic equations. We extend several classical theorems from complex analysis to solutions of any weakly hypoelliptic equation: the Montel theorem providing convergent subsequences, the Vitali theorem ensuring convergence of a given sequence, and Riemann's first removable singularity theorem. In the case of constant coefficients, we show that Liouville's theorem holds, any bounded solution must be constant, and any -solution must vanish.