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  "body": "\n## Statement\n\nLet $U \\subseteq \\R^n$ be open, and let\n$$L u = \\sum_{i,j=1}^{n} a_{ij}(x) \\, \\partial_i \\partial_j u + \\sum_{i=1}^{n} b_i(x) \\, \\partial_i u$$\nhave bounded coefficients on $U$, with $a_{ij} = a_{ji}$ and $\\sum_{i,j} a_{ij}(x) \\xi_i \\xi_j \\ge \\theta \\abs{\\xi}^2$ for some $\\theta \u003e 0$ and all $x \\in U$, $\\xi \\in \\R^n$.\nLet $x_0 \\in \\partial U$, and assume the interior ball condition at $x_0$: there is an open ball $B \\subseteq U$ with $x_0 \\in \\partial B$.\nLet $\\nu$ be the outer unit normal of $B$ at $x_0$.\nLet $u$ be twice continuously differentiable on $U$ and continuous at $x_0$ on $U \\cup \\set{x_0}$, with $L u \\ge 0$ in $U$ and $u(x) \u003c u(x_0)$ for all $x \\in U$.\nIf the one-sided derivative $\\frac{\\partial u}{\\partial \\nu}(x_0) = \\lim_{t \\to 0^+} \\frac{u(x_0) - u(x_0 - t \\nu)}{t}$ exists, then\n$$\\frac{\\partial u}{\\partial \\nu}(x_0) \u003e 0.$$\n\n## Remarks\n\nProof idea: let $B$ have centre $y$ and radius $R$, and set $v(x) = e^{-\\lambda \\abs{x - y}^2} - e^{-\\lambda R^2}$.\nFor $\\lambda$ large, $L v \u003e 0$ on the annulus $A = \\set{R/2 \u003c \\abs{x - y} \u003c R}$, and $v = 0$ on the outer sphere.\nFor small $\\varepsilon \u003e 0$, the function $u - u(x_0) + \\varepsilon v$ satisfies $L(\\cdot) \\ge 0$ in $A$ and is at most $0$ on $\\partial A$, so it is at most $0$ in $A$ by the weak [[maximum principle|D2JF19HF91]].\nAt $x_0$ this gives $\\frac{\\partial u}{\\partial \\nu}(x_0) \\ge -\\varepsilon \\frac{\\partial v}{\\partial \\nu}(x_0) = 2 \\varepsilon \\lambda R e^{-\\lambda R^2} \u003e 0$.\n\nThe interior ball condition is needed.\nLet $U$ be the part of the unit disc in the first quadrant, $x_0 = 0$ and $u(x, y) = -xy$.\nThen $\\Delta u = 0$ and $u \u003c 0 = u(x_0)$ in $U$, but $\\nabla u(0) = 0$, so every directional derivative at $0$ vanishes.\nAt the corner $0$ there is no ball inside $U$ touching $0$.\n\n## Explanations\n\n### L1 human\n\nAt a boundary point where a solution of an elliptic equation reaches a strict maximum, the solution cannot approach that value flatly.\nIt must rise towards the boundary with a strictly positive slope in the normal direction.\nFor a temperature in equilibrium, the hottest boundary point actually sends heat into the region.\n\n### L2 human\n\nThe Hopf lemma is the step that turns the weak maximum principle into the strong one.\nIf an interior maximum occurred, one would find a small ball touching the set where the maximum is attained.\nThe lemma would then give a nonzero gradient at a maximum point, which is impossible.\nIt also gives uniqueness up to constants for the Neumann problem, and it is a basic tool in the moving plane method for proving symmetry of solutions.\nThe smoothness of the boundary in the form of the interior ball condition holds, for example, for every bounded open set with $C^2$ boundary.\n",
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