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  "body": "\n## Definition\n\nLet $X$ and $Y$ be normed spaces of functions, for example [[Banach spaces|06GZ956657]], and let $\\mathcal{F}$ be a class of data $f \\in Y$.\nConsider a [[partial differential equation|GC5BDZ6QBC]] or boundary value problem $A(u) = f$.\nLet $C \\colon [0, \\infty) \\to [0, \\infty)$ be a function, often linear, that does not depend on $u$.\nAn *a priori estimate* for this problem is an inequality $\\norm{u}_X \\le C(\\norm{f}_Y)$.\nIt must hold for every solution $u \\in X$ of $A(u) = f$ with $f \\in \\mathcal{F}$.\nThe estimate assumes that $u$ is a solution in $X$; it asserts nothing about the existence of $u$.\n\n## Remarks\n\nExample: let $\\Omega \\subseteq \\R^n$ be bounded and open, $f \\in L^2(\\Omega)$, and let $u \\in H^1_0(\\Omega)$ be a weak solution of $-\\Delta u = f$.\nTesting with $u$ gives $\\int_\\Omega \\abs{\\nabla u}^2 = \\int_\\Omega f u \\le \\norm{f}_{L^2} \\norm{u}_{L^2}$.\nBy the [[Poincaré inequality|JS2CCR3RNS]], $\\norm{u}_{L^2} \\le C_P \\norm{\\nabla u}_{L^2}$, hence $\\norm{\\nabla u}_{L^2} \\le C_P \\norm{f}_{L^2}$.\nThe constant depends only on $\\Omega$.\n\nNon-example: a bound $\\norm{u}_X \\le C(u)$ with a constant that depends on the particular solution is not an a priori estimate.\nIt gives no uniform control over a family of solutions.\n\nThe [[Schauder estimates|HSEKZMAH1V]] and [[Calderón–Zygmund estimates|RYQJH4RYB4]] are a priori estimates in Hölder and $L^p$ norms.\nCombined with the Leray–Schauder fixed point method or the method of continuity, a priori estimates yield existence.\n",
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