FoCM manuscript No. (will be inserted by the editor) Overdetermined Elliptic Systems Katsiaryna Krupchyk1, Werner M. Seiler2, Jukka Tuomela1 1 Dept. of Mathematics, University of Joensuu, Finland, e-mail:
[email protected],
[email protected] 2 Interdisciplinary Centre for Scientific Computing, Universitat¨ Heidelberg, 69120 Heidel- berg, Germany, e-mail:
[email protected] Abstract We consider linear overdetermined systems of partial differential equa- tions. 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. Fur- thermore, 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 one with only one unknown function preserves ellipticity. Key words Overdetermined system, partial differential equation, symbol, ellip- tic system, completion, involution 1 Introduction The definition of ellipticity for general overdetermined systems is quite rarely found in the literature, one accessible exception being the encyclopaedia article [15, Def. 2.1]. Without the general definition one may encounter conceptual prob- lems already in very simple situations. For instance, consider the transformation of the two-dimensional Laplace equation uxx + uyy = 0 to the first order system (this is discussed in the recent textbook [42, Example 2.10]): ux = v , uy = w , vx + wy = 0 . The transformed system is not elliptic, although it is obviously equivalent to La- place’s equation.