Differential and Integral Equations

When is a given set of PDEs part of an elliptic system?

Michael Renardy

Full-text: Access denied (no subscription detected)

We're sorry, but we are unable to provide you with the full text of this article because we are not able to identify you as a subscriber. If you have a personal subscription to this journal, then please login. If you are already logged in, then you may need to update your profile to register your subscription. Read more about accessing full-text


We investigate the following question: Given a set of $k$ partial differential equations for $m$ unknowns, where $k < m$, can we find $m-k$ additional equations in such a way that the full set of equations forms an elliptic system? We formulate a maximal rank condition which is obviously necessary. In general, however, the maximal rank condition is sufficient only if we allow the introduction of additional variables, not just additional equations. In particular, the equation ${\rm div}\,u=0$ can be part of an elliptic system for the components of the vector field $u$ only if the space dimension is 1,2,4, or 8.

Article information

Differential Integral Equations, Volume 18, Number 2 (2005), 233-239.

First available in Project Euclid: 21 December 2012

Permanent link to this document

Mathematical Reviews number (MathSciNet)

Zentralblatt MATH identifier

Primary: 35J45


Renardy, Michael. When is a given set of PDEs part of an elliptic system?. Differential Integral Equations 18 (2005), no. 2, 233--239. https://projecteuclid.org/euclid.die/1356060231

Export citation