pp 43-44, Sobolev

*Characteristics:* We come at last to another central part of
the course [in many ways
today's lecture is the keystone of the whole course]. We saw that we
could set one of the
coefficients to zero; and if the equation was hyperbolic, this gave
the extra benefit of setting
*two* of them to zero. In general, we will be able to set
*one* of the coefficients to
zero with
relative ease; this section of Sobolev gives a geometric picture (in
words) of this process, and
thus gives one definition of a *characteristic surface*. In
general, changing to we have

we assume that near we have some actual dependence on the
. Sobolev
puts , which is true if any one of
them is nonzero. This is
geometrically equivalent to saying that for all small **c**
fills space around .
Sobolev shows is a condition of the surface
, and has nothing to do with

**Definition**: A surface is a
*characteristic* if
. (plus other `niceness' conditions).

Two deductions and we are done for the day:

- If our PDE is elliptic, it has no real characteristic surface, because is a positive definite quadratic form, and so .
- If we are on a characteristic surface, then the PDE itself
forces a relation between
**u**on and on . This means we cannot specify boundary conditions for both**u**and on a characteristic.

Wed Feb 4 14:46:04 EST 1998