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  "body": "\n## Statement\n\nLet $n \\ge 1$ and let $\\omega_n$ be the volume of the unit ball of $\\R^n$.\nLet $u$ be [[harmonic|0DEB0JMN0W]] on an open set containing the closed ball $\\overline{B}(x_0, r)$.\n\n1. The [[gradient|0ABK9V9Y64]] of $u$ at the centre satisfies\n$$\\abs{\\nabla u(x_0)} \\le \\frac{n}{r} \\, \\max_{\\overline{B}(x_0, r)} \\abs{u}.$$\n2. For every multi-index $\\alpha$ of order $k = \\abs{\\alpha}$,\n$$\\abs{\\partial^\\alpha u(x_0)} \\le \\frac{C_k}{r^{n+k}} \\int_{B(x_0, r)} \\abs{u(y)} \\, dy, \\qquad C_0 = \\frac{1}{\\omega_n}, \\quad C_k = \\frac{(2^{n+1} n k)^k}{\\omega_n} \\ (k \\ge 1).$$\n\n## Remarks\n\nProof idea for part 1: each partial derivative $\\partial_i u$ is harmonic, so by the [[mean value property|0HYPR64JM2]] over the ball and the [[divergence theorem|0JX86PSSA9]],\n$$\\partial_i u(x_0) = \\frac{1}{\\omega_n r^n} \\int_{B(x_0, r)} \\partial_i u \\, dy = \\frac{1}{\\omega_n r^n} \\int_{\\partial B(x_0, r)} u \\, \\nu_i \\, dS.$$\nThe last integral is at most $\\max \\abs{u}$ times the area $n \\omega_n r^{n-1}$ of the sphere, which gives $\\abs{\\partial_i u(x_0)} \\le \\frac{n}{r} \\max \\abs{u}$.\nThe same argument for the derivative in the direction of $\\nabla u(x_0)$ gives the bound for $\\abs{\\nabla u(x_0)}$.\nPart 2 follows by induction on $k$, applying the same argument on balls of radius $r/k$.\nExample: $u(x, y) = x$ on the unit disc has $\\abs{\\nabla u} = 1$ and $\\max \\abs{u} = 1$, within the bound $2$ of part 1.\nThe estimates are interior: they control derivatives at the centre by the size of $u$ on the ball, and they fail near the boundary of the domain.\nFor instance $u_m(x, y) = \\operatorname{Re}(x + iy)^m$ on the unit disc has $\\abs{u_m} \\le 1$ but $\\abs{\\nabla u_m(1, 0)} = m$.\n\n## Explanations\n\n### L1 human\n\nA harmonic function cannot change quickly unless it is large.\nIf it stays between $-M$ and $M$ on a ball of radius $r$, its slope at the centre is at most $n M / r$.\nBig balls therefore force small slopes, and this is why bounded harmonic functions on the whole space are constant.\n\n### L2 human\n\nThese estimates give Liouville's theorem, the real analyticity of harmonic functions, and the compactness of bounded families of harmonic functions through the Arzelà–Ascoli theorem.\nThey are the model for interior Schauder and Sobolev estimates for general elliptic equations.\nFor nonnegative harmonic functions there is a sharper form, $\\abs{\\nabla u(x_0)} \\le \\frac{n}{r} u(x_0)$, which leads to Harnack's inequality.\n",
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