If a PDE has a unique classical solution, must it have a unique viscosity solution?


The particular problem I am interested in is parabolic, but I would be interested in the general case.


A short answer would be good. An answer with references would be great!



If a PDE has a unique classical solution, must it have a unique viscosity solution?
The answer in general is no, because viscosity solutions are not suitable for every problem.


The viscosity solution idea based on an idea of solution comparison. If you look at the user's guide, Crandall-Ishii-Lions 1992, you see that for a problem of the form
$$
F(x,u,Du,D^2u)=0
$$
what they call the fundamental assumption is that
$$
F(x,r,p,X)\leq F(x,s,p,Y) \mbox{ whenever } r\leq s \mbox{ and } Y\leq X.
$$
There must be a way to define subsolutions and supersolutions. You cannot do this for an arbitrary PDE. For example, if you consider an oscillatory problem, e.g. Maxwell's Equations, that seems very difficult.


On the other hand, if your problem is of the correct type, then yes, of course.
A basic references to know what correct type means are the User's guide given above. I would also look at Guy Barles' book (1994), where he gives a number of conditions grouped under the idea of 'Natural conditions' (e.g., coercivity, convexity etc..) where uniqueness holds for parabolic problems. 


