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  "body": "\n## Summary\n\nAn a priori estimate is a bound on a solution of a differential equation in terms of the data.\nIt is proved under the assumption that the solution already exists and is regular enough.\nIt is proved before, and independently of, any existence result.\nA priori estimates are the main input of existence proofs by continuity, by fixed points, or by compactness, and they also give uniqueness and stability.\n\n## Explanations\n\n### L1 human\n\nSuppose one wants to show that an equation has a solution, but cannot write it down.\nA common strategy is to first assume a solution exists and show it cannot be too large.\nThis bound is the a priori estimate: \"a priori\" because it comes before knowing that a solution exists.\nFor example, a solution of $-u'' = f$ on $(0, 1)$ with $u(0) = u(1) = 0$ always satisfies $\\max \\abs{u} \\le \\frac{1}{8} \\max \\abs{f}$.\nWith such bounds in hand, approximate solutions stay under control and a limit gives a true solution.\n",
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