jump discontinuity proof for PDE
So here is the problem statement:
In addition, the following "hint" is given:
Consider the equation (6.12)
solution of (6.12) in each of two regions separated by a curve
be continuous, but
have a jump discontinuity along the curve. Prove that
and hence that the curve is characteristic.
I believe denote the limits of from the right and left, respectively, and similarly for .
Hint: By (6.12)
are continuous on the curve.
This is all from Fritz John's Partial Differential Equations, exercise 1.6.3, p19.
I'm pretty lost on this one. Any help would be much appreciated. Thanks !
Re: jump discontinuity proof for PDE
I've been working on this for a bit, and I think I have an idea. However, I still need some help making the idea work.
According to the textbook (Fritz John, p17), if we choose satisfying , then we have the following "conservation law":
and therefore if we choose then
Now, if we choose and then this will satisfy the conservation law, giving us
If I can show that this implies
then it will follow that
and taking limits as will give us the desired result .
But the problem is, I don't know how to show that (1) implies (2). Any suggestions ?
Thanks again guys !