I would like an example where the maximum principle fails in a bounded smooth domain $\Omega$ where one has a solution which is not $C^2(\Omega)$ to $Lu=0$ where $L$ is elliptic and linear. This obviously must rely on the coefficients being discontiuous for the elliptic operator since otherwise one can do interior regularity estimates. All of the examples I have tried to come up with turn out to not actually be weak solutions so I'm stuck on this. Perhaps maximum principles extend to the nonsmooth setting? Any insight on this would be appreciated.
Yes indeed, the maximum principle extends to the nonsmooth setting. I am not sure that there is a complete theory, because so many situations can occur. But at least let me mention the following situations.



Maximum principle, in general, can be applied in the viscosity solutions setting (in this case, the viscosity solution is only continuous). You can have a look at the excellent paper by CrandallIshiiLions as following: http://arxiv.org/abs/math/9207212 

