## Differential and Integral Equations

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

Michael Renardy

#### Abstract

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

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

Dates
First available in Project Euclid: 21 December 2012

Permanent link to this document
https://projecteuclid.org/euclid.die/1356060231

Mathematical Reviews number (MathSciNet)
MR2106104

Zentralblatt MATH identifier
1212.35106

Subjects
Primary: 35J45

#### Citation

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