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  "body": "\n## Statement\n\nLet $n \\ge 1$, let $0 \u003c \\alpha \u003c 1$, and let $U \\subseteq \\R^n$ be a bounded open set.\nLet\n$$L u = \\sum_{i,j=1}^{n} a_{ij}(x) \\, \\partial_i \\partial_j u + \\sum_{i=1}^{n} b_i(x) \\, \\partial_i u + c(x) \\, u,$$\nwhere $a_{ij} = a_{ji}$, $b_i$ and $c$ are functions on $U$.\nLet $\\lambda \u003e 0$ and $\\Lambda$ be constants with $\\sum_{i,j} a_{ij}(x) \\xi_i \\xi_j \\ge \\lambda \\abs{\\xi}^2$ for all $x \\in U$ and $\\xi \\in \\R^n$.\nAssume that every coefficient has $C^{0,\\alpha}(\\overline{U})$ norm at most $\\Lambda$.\nHere $C^{k,\\alpha}$ denotes the [[Hölder space|P101AX1ZSC]] of functions whose derivatives up to order $k$ are bounded and [[Hölder continuous|012PBA0FMD]] with exponent $\\alpha$.\nIts norm is written $\\norm{\\cdot}_{C^{k,\\alpha}}$.\n\n1. (Interior estimate.) Let $u \\in C^{2,\\alpha}(U)$ be bounded with $f = L u \\in C^{0,\\alpha}(\\overline{U})$, and let $V$ be an open set whose closure is a compact subset of $U$. Then\n$$\\norm{u}_{C^{2,\\alpha}(\\overline{V})} \\le C \\big( \\sup_U \\abs{u} + \\norm{f}_{C^{0,\\alpha}(\\overline{U})} \\big),$$\nwhere $C$ depends only on $n$, $\\alpha$, $\\lambda$, $\\Lambda$, $U$ and $V$.\n2. (Global estimate.) Assume moreover that $U$ has $C^{2,\\alpha}$ boundary. Let $u \\in C^{2,\\alpha}(\\overline{U})$, let $f = L u$, and let $\\varphi \\in C^{2,\\alpha}(\\overline{U})$ with $u = \\varphi$ on $\\partial U$. Then\n$$\\norm{u}_{C^{2,\\alpha}(\\overline{U})} \\le C \\big( \\sup_U \\abs{u} + \\norm{\\varphi}_{C^{2,\\alpha}(\\overline{U})} + \\norm{f}_{C^{0,\\alpha}(\\overline{U})} \\big),$$\nwhere $C$ depends only on $n$, $\\alpha$, $\\lambda$, $\\Lambda$ and $U$.\n\n## Remarks\n\nProof idea: first treat the Laplacian, where $u$ is given by convolution of $f$ with the fundamental solution.\nPotential estimates show that the second derivatives of the Newtonian potential of a $C^{0,\\alpha}$ function are $C^{0,\\alpha}$.\nA linear change of variables gives the same bound for operators with constant coefficients.\nFor variable coefficients one freezes them at a point.\nOn a small ball, $L$ is close to a constant-coefficient operator, and the error is absorbed using the smallness of the Hölder seminorm of the coefficients on small balls.\nThe global estimate follows by flattening the boundary locally and treating half-balls in the same way.\n\nExample: for $L = \\Delta$, the estimate says that a harmonic function is controlled in $C^{2,\\alpha}$ on interior sets by its supremum.\nIt also says that a solution of Poisson's equation $\\Delta u = f$ with $f \\in C^{0,\\alpha}$ has second derivatives in $C^{0,\\alpha}$.\n\nThe restriction $0 \u003c \\alpha \u003c 1$ is needed.\nFor $\\alpha = 0$ the estimate fails: there are continuous $f$ for which $\\Delta u = f$ has no solution with continuous second derivatives.\nFor $\\alpha = 1$ it also fails: a Lipschitz $f$ need not give $u$ with Lipschitz second derivatives.\n\nTogether with the method of continuity, the global estimate gives existence of $C^{2,\\alpha}(\\overline{U})$ solutions of the Dirichlet problem for $L$ with $c \\le 0$, starting from the case of the Laplacian.\n\n## Explanations\n\n### L1 human\n\nThe Schauder estimates say that solutions of elliptic equations are two derivatives smoother than the right-hand side, when smoothness is measured with Hölder norms.\nIf the data are slightly better than continuous, the second derivatives of the solution are slightly better than continuous too, with a bound on how much.\nThis is a quantitative form of the idea that elliptic equations smooth things out.\n\n### L2 human\n\nThe Schauder estimates are a priori estimates: they bound the $C^{2,\\alpha}$ norm of a solution that is already known to exist.\nCombined with the maximum principle, which gives uniqueness and bounds $\\sup \\abs{u}$ when $c \\le 0$, they give existence of classical solutions through the method of continuity or the Leray–Schauder theory.\nApplied to derivatives of $u$, they give higher regularity: if the coefficients and $f$ are in $C^{k,\\alpha}$, then $u$ is in $C^{k+2,\\alpha}$ inside $U$.\nThey are the Hölder counterpart of the $L^p$ theory of Calderón and Zygmund.\nThey are also a key step in nonlinear problems such as Hilbert's nineteenth problem, once the De Giorgi–Nash–Moser theorem has given Hölder continuity of the coefficients.\n",
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