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  "body": "\n## Statement\n\nLet $U \\subseteq \\R^n$ be a bounded open set and let $m \\ge 0$ be an integer.\nLet\n$$L u = -\\sum_{i,j=1}^{n} \\partial_j \\big( a_{ij}(x) \\, \\partial_i u \\big) + \\sum_{i=1}^{n} b_i(x) \\, \\partial_i u + c(x) \\, u,$$\nwhere $a_{ij} = a_{ji}$, $b_i$ and $c$ are in $C^{m+1}(\\overline{U})$.\nAssume that $L$ is uniformly [[elliptic|XY8GEQSWPR]]: there is $\\theta \u003e 0$ with $\\sum_{i,j} a_{ij}(x) \\xi_i \\xi_j \\ge \\theta \\abs{\\xi}^2$ for all $x \\in U$ and $\\xi \\in \\R^n$.\nLet $f \\in H^m(U)$, and let $u \\in H^1(U)$ be a [[weak solution|SJZX81A783]] of $L u = f$ in $U$.\nHere $H^k(U) = W^{k,2}(U)$ is the [[Sobolev space|M4CEZ5K9HP]] of functions on $U$ whose [[weak derivatives|C3EC148VZD]] up to order $k$ lie in $L^2(U)$.\n\n1. Let $V$ be an open set whose closure is a compact subset of $U$. Then $u \\in H^{m+2}(V)$, and\n$$\\norm{u}_{H^{m+2}(V)} \\le C \\big( \\norm{f}_{H^m(U)} + \\norm{u}_{L^2(U)} \\big),$$\nwhere $C$ depends only on $m$, $U$, $V$ and the coefficients of $L$.\n2. If the coefficients of $L$ and $f$ are [[smooth|0V33BNKZ9S]] on $\\overline{U}$, then $u$ agrees almost everywhere with a smooth function on $U$.\n\n## Remarks\n\nProof idea for $m = 0$: multiply the weak form by a cut-off function and test it with difference quotients $D_k^h u$ of $u$ in the direction $e_k$.\nUniform ellipticity bounds $\\norm{D_k^h \\nabla u}_{L^2}$ on $V$ independently of $h$, and a uniform bound on difference quotients implies that the weak derivative $\\partial_k \\nabla u$ exists in $L^2(V)$.\nThe case of general $m$ follows by induction, since a derivative of $u$ solves an equation of the same form with a right-hand side in $H^{m-1}$.\nPart 2 follows from part 1 for all $m$ together with the Sobolev embedding theorem, which puts $H^{k}$ inside $C^j$ for $k \u003e j + n/2$.\n\nExample: a weak solution $u \\in H^1(U)$ of $\\Delta u = 0$ is smooth; this is Weyl's lemma, and $u$ is then a classical harmonic function.\n\nEllipticity is needed.\nFor the wave operator $\\partial_t^2 - \\partial_x^2$ in the plane, $u(t, x) = \\abs{x - t}$ is a weak solution of $\\partial_t^2 u - \\partial_x^2 u = 0$ that is not differentiable on the line $x = t$.\n\nThe regularity is local: without conditions on the boundary of $U$ and on the boundary values, $u$ need not be in $H^{m+2}(U)$ up to the boundary.\nFor the Dirichlet problem with $u \\in H^1_0(U)$ and $\\partial U$ of class $C^{m+2}$, the same conclusion holds on all of $U$.\n\n## Explanations\n\n### L1 human\n\nElliptic regularity says that solutions of elliptic equations are as smooth as the data allow, and in fact two derivatives smoother than the right-hand side.\nOne may first find a rough solution, defined only through integrals against test functions.\nThe theorem then shows that this rough solution is automatically smooth when the coefficients and the right-hand side are smooth.\nFor Laplace's equation, every weak solution is a genuine smooth harmonic function.\n\n### L2 human\n\nThe standard way to solve a linear elliptic problem has two steps.\nFirst one gets a weak solution in $H^1$ from the Lax–Milgram theorem or the Riesz representation theorem.\nThen elliptic regularity upgrades it to $H^{m+2}$, and with Sobolev embeddings to a classical solution.\nThe same pattern holds in other function spaces: the Calderón–Zygmund estimates give the $W^{2,p}$ version for $1 \u003c p \u003c \\infty$, and the Schauder estimates give the $C^{2,\\alpha}$ version.\nHyperbolic operators have no such property, since their solutions propagate singularities of the data.\n",
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