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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 non-smooth setting? Any insight on this would be appreciated.

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Yes indeed, the maximum principle extends to the non-smooth 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.

  1. In Hopf's maximum principle, one only needs that $u$ admits second order derivatives (this is less than ${\mathcal C}^2$) and the coefficients of $L$ be bounded.
  2. In the variational case, that if $L$ is self-adjoint, the $H^1$ solution is unique and minimizes a functional $\int (Lu,u)dx$. Using the fact that if $v$ is in the Sobolev space $H^1$, then the absolute value $|v|$ is in this space too, one again proves a maximum principle.
  3. The previous item covers the one-dimensional case, because then second order operator can always be multiplied by an appropriate weight in such a way that they become self-adjoint. This is the reason why the spectrum of a one-D second-order operator is real (Liouville's theory).
  4. If $L$ is elliptic with smooth coefficients, and if $u$ is a distribution such that $Lu\in{\mathcal C}^\infty$, then $u\in{\mathcal C}^\infty$. Therefore the maximum principle is valid.
  5. The modern theory of second-order elliptic (not necessarily linear) equations is based upon the maximum principle. One declares that $u\in{\mathcal C}^0$ is a solution if, given an arbitrary point, there exists a super-solution (resp. a sub-solution) $\phi_\pm$ such that $\phi_\pm(x)=u(x)$, and $\phi_+\le u\le \phi_-$ otherwise.
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Thanks for all of the nice examples and facts Denis! I'll use them to think a bit more about this. –  Dorian Sep 19 '10 at 19:52
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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 Crandall-Ishii-Lions as following: http://arxiv.org/abs/math/9207212

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