 Hopf lemma

For $L=a^{ij}{D}_{ij}+b^i{D}_i$ with bounded coefficients and uniform ellipticity near a boundary point $y$ satisfying the interior sphere condition, suppose $u\in C^2(\Omega )\cap C^0(\Omega \cup \{y\})$ has $Lu\ge 0$ and $u(x)<u(y)$ in a tangent open ball. With $\mathbf{n}$ the outward unit normal of that ball, ${\mathrm{liminf}}_{t\downarrow 0}[u(y)-u(y-t\mathbf{n})]/t>0$. Consequently the outward normal derivative is positive whenever it exists. The given boundary continuity alone does not assert that the derivative exists. The minimum version follows by applying this to $-u$; operators with a zeroth-order term require additional sign hypotheses.