Overdetermined Elliptic Systems

Authors:
Katsiaryna Krupchyk;Werner M. Seiler;Jukka Tuomela
Affiliations:
Department of Mathematics, University of Joensuu, Finland;Interdisciplinary Centre for Scientific Computing, Universitat Heidelberg, 69120 Heidelberg, Germany;Department of Mathematics, University of Joensuu, Finland
Venue:
Foundations of Computational Mathematics
Year:
2006

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Abstract

We consider linear overdetermined systems of partial differential equations. We show that the introduction of weights classically used for the definition of ellipticity is not necessary, as any system that is elliptic with respect to some weights becomes elliptic without weights during its completion to involution. Furthermore, it turns out that there are systems which are not elliptic for any choice of weights but whose involutive form is nevertheless elliptic. We also show that reducing the given system to lower order or to an equivalent system with only one unknown function preserves ellipticity.