the Hopf boundary-point lemma
/ HOPF /
The maximum principle tells you a harmonic function's largest value sits on the boundary. The Hopf lemma asks a finer question: as you walk out to that boundary maximum, how is the function behaving? The answer is that it must be strictly increasing as it reaches the peak — it cannot flatten out and arrive with zero slope. The function leans into the boundary at its hottest edge point, with a genuinely negative inward-pointing derivative.
Precisely: suppose u is subharmonic (or solves a uniformly elliptic equation), the domain satisfies an interior-ball condition at a boundary point P (a ball inside the domain touches the boundary at P), u attains a strict maximum M at P, and u < M inside. Then the outward normal derivative at P is strictly positive: the slope of u in the direction pointing OUT of the domain is bounded away from zero, not merely nonnegative. Equivalently the inward normal derivative is strictly negative. The proof slips a small comparison function (built from something like e^(-alpha r^2) minus a constant, on an annulus inside the touching ball) under u; this barrier has a known strictly positive outward slope at P, and since u sits above it and they agree at P, u must descend at least as steeply.
Why care about a derivative at one point? Because it is the missing ingredient for Neumann and Robin uniqueness and for symmetry. If two solutions of a Neumann problem differed, their difference would attain an interior-or-boundary extremum; the strong maximum principle handles interior extrema, and the Hopf lemma handles a boundary extremum by forcing a nonzero normal derivative there — contradicting the prescribed zero (or matched) Neumann data. The lemma is also the engine that starts the moving-plane method, where you need strict monotonicity at the reflecting boundary to get the plane moving.
On a disk, a harmonic function whose maximum is attained at a single boundary point cannot approach that point with a flat profile: its radial derivative pointing outward at the peak is strictly positive. You can rule out a horizontal tangent at the hottest edge point without computing the solution.
At a boundary maximum the function arrives with a strictly nonzero inward slope.
It needs an interior-ball condition (the domain must be smooth enough at P to fit a ball tangent there); at a sharp inward corner the conclusion can fail. The lemma gives strict positivity of the normal derivative, not just nonnegativity — that strictness is exactly what powers the uniqueness and symmetry arguments.