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- Title
Some Properties of Solutions to Weakly Hypoelliptic Equations.
- Authors
Bär, Christian
- 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 LP-solution must vanish.
- Subjects
NUMERICAL analysis; HYPOELLIPTIC differential equations; LINEAR operators; MATRICES (Mathematics); COEFFICIENTS (Statistics); PARABOLIC differential equations; STOCHASTIC convergence; MATHEMATICAL bounds
- Publication
International Journal of Differential Equations, 2013, p1
- ISSN
1687-9643
- Publication type
Article
- DOI
10.1155/2013/526390