My current research took me to the realm of PDE's (which for the long time used to be terra incognita for me as I'am a probabilist). Equations that I'am working with are mostly of second order or Hamilton-Jacobi equations. It's no suprise that I'am dealing with various notions of weak solutions (viscosity solutions mostly) without a hope to get classical solution. So here is my question:

is it possible that equation $F(x,u,Du,D^2u)=0$ which is nondegenerate, nonsingular, with smooth $F$ has some sort of weak solutions (viscosity or other) but doesn't have classical solutions.

I saw many results on regularity of solutions in various settings but none of encountered theorems even seem to get close to answer my question without awkward assumptions. I'd be glad if someone who is familiar with PDE's could tell me if such a general theorems exists nowadays.

Elliptic Partial Differential Equations, you have many useful estimate as Harnack inequality or $W^{2,p}$-estimate. – Paul Nov 23 '11 at 9:57