Overdetermined Elliptic Systems

Co-author(s): Katsiaryna Krupchyk and Jukka Tuomela

Reference: Foundations of Computational Mathematics, 6 (2006) 309-351

Description: We discuss the notion of ellipticity for overdetermined systems of differential equations. In particular, the reduction to first order of a higher order system always leads to an overdetermined system and the classical definition of ellipticity fails, if one does not include all integrability conditions. As a remedy a weighted principal symbol has been introduced by Douglis and Nirenberg (ellipticity in the sense of Petrovsky is a special case of this construction). We show that this approach is neither necessary nor sufficient for deciding ellipticity. By contrast, it always suffice to complete first to an involutive system and then check for ellipticity. Appropriately chosen weights partially simulate the effect of a completion and hence in many (but not all) cases allow us to decide ellipticity.

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