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  "body": "\n## Statement\n\nLet $\\Omega \\subseteq \\R^n$ be open and let $0 \u003c \\lambda \\le \\Lambda$.\nLet $a_{ij} \\colon \\Omega \\to \\R$, for $1 \\le i, j \\le n$, be measurable functions with $\\abs{a_{ij}(x)} \\le \\Lambda$ and\n$$\\sum_{i,j=1}^n a_{ij}(x) \\xi_i \\xi_j \\ge \\lambda \\abs{\\xi}^2 \\quad \\text{for all } x \\in \\Omega \\text{ and } \\xi \\in \\R^n.$$\nLet $u \\in W^{1,2}_{\\mathrm{loc}}(\\Omega)$ be a [[weak solution|SJZX81A783]] of $\\sum_{i,j} \\partial_i (a_{ij} \\partial_j u) = 0$ in $\\Omega$, where $W^{1,2}_{\\mathrm{loc}}$ is a local [[Sobolev space|M4CEZ5K9HP]].\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. $u$ is essentially bounded on $B_r(x_0)$ and $\\displaystyle \\sup_{B_r(x_0)} \\abs{u} \\le C \\Big( r^{-n} \\int_{B_{2r}(x_0)} u^2 \\, dx \\Big)^{1/2}$.\n2. After change on a null set, $u$ is continuous, and $\\abs{u(x) - u(y)} \\le C \\big(\\abs{x - y}/r\\big)^{\\alpha} \\sup_{B_{2r}(x_0)} \\abs{u}$ for all $x, y \\in B_r(x_0)$.\n\n## Remarks\n\nThe point is that the coefficients $a_{ij}$ are only measurable, so no smoothness of the equation is passed on to $u$; the [[Hölder continuity|012PBA0FMD]] comes from ellipticity alone.\nDe Giorgi and Nash proved the result independently; De Giorgi used truncations at many levels and an iteration on the measure of super-level sets.\nMoser later gave a proof by iterating reverse Hölder inequalities for powers of $u$, and he obtained a Harnack inequality on the way.\nThe theorem solves Hilbert's nineteenth problem.\nLet $F$ be smooth and uniformly convex, and let $u$ minimize $\\int F(\\nabla u)\\,dx$ in $W^{1,2}$.\nThen the derivatives $\\partial_k u$ solve an equation of the form above, so they are Hölder continuous.\nThe Schauder estimates then make $u$ smooth.\nThe result fails for systems: De Giorgi gave an example of an elliptic system with bounded measurable coefficients with a discontinuous weak solution.\n\n## Explanations\n\n### L1 human\n\nThe theorem looks at the simplest kind of [[elliptic equation|B4TTNHSTRX]], but with very rough coefficients that may jump around.\nEven so, every solution is continuous, and in fact Hölder continuous, with bounds that depend only on how elliptic the equation is.\nThis gives a floor of smoothness that rough data cannot break.\n\n### L2 human\n\nThis is the key step of regularity theory for nonlinear elliptic equations in divergence form.\nDifferentiating a nonlinear equation gives a linear equation for the derivatives whose coefficients are only known to be bounded and elliptic; this theorem then makes the derivatives Hölder continuous.\nAfter that, the Schauder estimates bootstrap the regularity to smoothness.\nThe parabolic version, also due to Nash and Moser, plays the same role for parabolic equations in divergence form.\nThe counterpart for equations in non-divergence form is the Krylov–Safonov theorem.\n",
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