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  "body": "\n## Statement\n\nLet $\\Omega \\subseteq \\R^n$ be open and let $0 \u003c \\lambda \\le \\Lambda$.\nLet $a_{ij} = a_{ji} \\colon \\Omega \\to \\R$ be measurable functions with\n$$\\lambda \\abs{\\xi}^2 \\le \\sum_{i,j=1}^n a_{ij}(x) \\xi_i \\xi_j \\le \\Lambda \\abs{\\xi}^2 \\quad \\text{for all } x \\in \\Omega \\text{ and } \\xi \\in \\R^n.$$\nLet $u \\in W^{2,n}_{\\mathrm{loc}}(\\Omega)$, a local [[Sobolev space|M4CEZ5K9HP]], satisfy $\\sum_{i,j} a_{ij} \\partial_i \\partial_j u = 0$ almost everywhere in $\\Omega$.\nThere are constants $C \u003e 0$ and $\\alpha \\in (0,1)$, depending only on $n$, $\\lambda$ and $\\Lambda$, such that for every ball $B_{2r}(x_0)$ with closure in $\\Omega$:\n\n1. (Harnack inequality) If $u \\ge 0$ in $B_{2r}(x_0)$, then $\\sup_{B_r(x_0)} u \\le C \\inf_{B_r(x_0)} u$.\n2. (Hölder estimate) $\\abs{u(x) - u(y)} \\le C \\big(\\abs{x-y}/r\\big)^{\\alpha} \\sup_{B_{2r}(x_0)} \\abs{u}$ for all $x, y \\in B_r(x_0)$.\n\n## Remarks\n\nThe equation is an [[elliptic equation|B4TTNHSTRX]] in non-divergence form and the coefficients are only measurable, so the energy methods of De Giorgi, Nash and Moser do not apply.\nThe proof uses the Aleksandrov–Bakelman–Pucci maximum principle to show that a supersolution which is large on a fixed fraction of a ball is bounded below on a smaller ball.\nA covering argument with dyadic cubes in the style of Calderón and Zygmund then turns this into a decay of the measure of level sets, which gives the Harnack inequality.\nThe [[Hölder continuity|012PBA0FMD]] estimate follows from the Harnack inequality applied to $\\sup u - u$ and $u - \\inf u$ on shrinking balls.\nFor $a_{ij} = \\delta_{ij}$ the first part is the classical [[Harnack inequality|YNH832D50W]] for harmonic functions.\n\n## Explanations\n\n### L1 human\n\nA non-negative solution of an elliptic equation cannot be huge at one point and tiny at a nearby point: its largest and smallest values on a ball are comparable.\nThis stays true even when the coefficients of the equation are very rough.\nFrom this one gets that solutions are Hölder continuous.\n\n### L2 human\n\nThis is the non-divergence analogue of the De Giorgi–Nash–Moser theorem.\nIt is the starting point of the regularity theory of fully nonlinear elliptic equations $F(D^2 u) = 0$: differences of solutions satisfy linear equations with measurable coefficients, so they are Hölder continuous.\nWith concavity of $F$ this leads to the Evans–Krylov theorem on $C^{2,\\alpha}$ regularity.\nCaffarelli extended the theory to viscosity solutions.\n",
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