Classification of linear PDE

Sobolev makes heavy going of this because linear algebra was not so
widely used in this day (it
really is a relatively recent phenomenon -- currently only the
British-trained fluid dynamicists
hold out against it (partly because tensor notation is so much more
economical)). The discussion
on pp 33-36 boils down to this (using the Einstein summation
convention, as a sop to the
tensorialists among you). Suppose our linear 2 order PDE
operator has leading terms

We may take symmetric because
for smooth **u**. Then we take
the eigendecomposition of **A**, namely

(orthogonal **Q** because **A** is symmetric, and diag is
real). Put diag except put
**1**'s when . Then

where

Then choose (a linear
change of variables) and
the
PDE is converted to its *canonical form*

Sylvester's law of inertia says that the number of positive 's,
negative 's, and zero
's, is invariant --- that is, they categorize the PDE.

n.b. these transformations are not unique.

Let **r** be the number of positive and let **s** be the number of
negative . Then the
triple categorizes the PDE with hyperbolic

(wave equation in 2D: type)

= elliptic

(Laplace's eqn)

= *parabolic* (normal)

(heat equation)

If the PDE has variable coefficients e.g. then the equation may change types in different regions. This will be important when we come to consider domains of influence and domains of dependence.

If, however, there are only two independent variables **x** and **y**,
then a linear 2 order
variable coefficient PDE can generically be brought to canonical form.

Wed Feb 4 14:46:04 EST 1998