International Journal of Differential Equations

Some Properties of Solutions to Weakly Hypoelliptic Equations

Christian Bär

Full-text: Open access

Abstract

A linear different operator L is called weakly hypoelliptic if any local solution u of Lu=0 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 Lp-solution must vanish.

Article information

Source
Int. J. Differ. Equ., Volume 2013 (2013), Article ID 526390, 8 pages.

Dates
Received: 21 May 2013
Accepted: 4 July 2013
First available in Project Euclid: 20 January 2017

Permanent link to this document
https://projecteuclid.org/euclid.ijde/1484881345

Digital Object Identifier
doi:10.1155/2013/526390

Mathematical Reviews number (MathSciNet)
MR3083295

Zentralblatt MATH identifier
1297.35087

Citation

Bär, Christian. Some Properties of Solutions to Weakly Hypoelliptic Equations. Int. J. Differ. Equ. 2013 (2013), Article ID 526390, 8 pages. doi:10.1155/2013/526390. https://projecteuclid.org/euclid.ijde/1484881345


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