= Hopf lemma
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{wiki}
= Hopf boundary point lemma
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{synonym}
For $L=a^{ij}D_{ij}+b^iD_i$ 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 $Lu\geq0$ and $u(x)<u(y)$ in a tangent <open ball>. With $\mathbf n$ the outward <unit normal> of that ball, $\liminf_{t\downarrow0}[u(y)-u(y-t\mathbf n)]/t>0$. 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.
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