In an introductory course on PDE's I got as a project to prove and present a version of Hopf's (boundary) lemma. Namely:
Let $\Omega \subset R^{d}$ be an non-empty open connected set with a twice differentiable boundary. Let $u\in C^{2}(\Omega) \cap > C^{1}(\bar{\Omega})$ be a subharmonic function ($\Delta u \geq 0)$. If there exists a $M\geq 0$ such that $u(x) < M$ for all $x\in \Omega$ and $u(x_{0}) = M$ for some $x_{0}$ on the boundary of $\Omega$ then the outward normal derivative at $x_{0}$ $(\nabla u(x_{0})\cdot n)$ is stricly positive.
Where can I find the original source of Hopf's lemma? I have looked trough Evans - Partial Differential Equations (second edition) and some notes written by professor John K. Hunter at UC Davis, but have been unable to make any progress.
