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arXiv · 1207.4036

Some properties of solutions to weakly hypoelliptic equations

Christian Baer
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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 cover all 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 L^p-solution must vanish.

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Christian Baer. 2013-08-01. Some properties of solutions to weakly hypoelliptic equations. https://doi.org/10.1155/2013%2F526390

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