Let $Lu=a^{ij}u_{ij} + b^i u_i$ be an elliptic operator of second order in a bounded domain $\Omega$. Assume that $a^{ij}$ is uniformly elliptic. Then it's well known that the following maximum principle holds:
Theorem: If $u \in C^2(\Omega)$, $Lu=0$, and $\limsup\limits_{x \to p} u(x) \le 0$ for every $p \in \partial \Omega$. Then $\text{sup}_\Omega u \le 0$.
My question is:
Question: Can the hypothesis "$\limsup\limits_{x \to p} u(x) \le 0$ for every $p \in \partial \Omega$" be weakened in any way ? For example, suppose that this inequality holds for all but one point on the boundary(I remember there's some literature on this case, but cann't find it any more), or holds for almost every point on the boundary.
