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
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)
= parabolic (normal)
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.