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Lecture 3, Section 4

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:

  1. If our PDE is elliptic, it has no real characteristic surface, because is a positive definite quadratic form, and so .
  2. 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.

Robert Corless
Wed Feb 4 14:46:04 EST 1998