Some properties of solutions to weakly hypoelliptic equations

  • A linear differential operator L is called weakly hypoelliptic if any local solution u of Lu = 0 is smooth. We allow for systems, i.e. the coefficients may be matrices, not necessarily of square size. This is a huge class of important operators which covers 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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Author:Christian Bär
Series (Serial Number):Preprints des Instituts für Mathematik der Universität Potsdam (1(2012)22)
Document Type:Preprint
Year of Completion:2012
Publishing Institution:Universität Potsdam
Release Date:2012/07/06
Tag:Hypoelliptic operators; Liouville theorem; Montel theorem; Vitali theorem; hypoelliptic estimate
Organizational units:Mathematisch-Naturwissenschaftliche Fakultät / Institut für Mathematik
Dewey Decimal Classification:5 Naturwissenschaften und Mathematik / 51 Mathematik / 510 Mathematik
MSC Classification:35-XX PARTIAL DIFFERENTIAL EQUATIONS / 35Bxx Qualitative properties of solutions / 35B53 Liouville theorems, Phragmén-Lindelöf theorems
35-XX PARTIAL DIFFERENTIAL EQUATIONS / 35Hxx Close-to-elliptic equations and systems / 35H10 Hypoelliptic equations
Collections:Universität Potsdam / Schriftenreihen / Preprints des Instituts für Mathematik der Universität Potsdam, ISSN 2193-6943 / 2012
Licence (German):License LogoKeine Nutzungslizenz vergeben - es gilt das deutsche Urheberrecht
Notes extern:RVK-Klassifikation: SI 990