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  "body": "\n## Definition\n\nLet $X$ and $Y$ be [[normed vector spaces|03J5RAKB30]], the space of data and the space of solutions.\nLet $D \\subseteq Y$ and let $A \\colon D \\to X$ be a map.\nThe problem of finding, for given $f \\in X$, some $u \\in D$ with $A(u) = f$ is *well posed* if:\n\n1. for every $f \\in X$ there is a solution $u \\in D$ of $A(u) = f$;\n2. for every $f \\in X$ this solution is unique;\n3. the solution map $X \\to Y$, $f \\mapsto u$, is [[continuous|0X3WGX95FE]].\n\nOtherwise the problem is *ill posed*.\n\n## Remarks\n\nThe choice of the spaces $X$ and $Y$ and of their norms is part of the problem.\nA problem can be well posed for one choice and ill posed for another.\nExample: the [[Dirichlet problem|8D56GQ4AG8]] for the Laplace equation on the closed unit disc $\\overline{B}$, with $X = C(\\partial B)$ and $Y = C(\\overline{B})$ both with the supremum norm.\nThe Poisson integral gives a solution for each boundary function $\\varphi$.\nIf $u$ and $v$ solve the problem with boundary values $\\varphi$ and $\\psi$, the [[maximum principle|D2JF19HF91]] applied to $\\pm(u - v)$ gives $\\max_{\\overline{B}} \\abs{u - v} \\le \\max_{\\partial B} \\abs{\\varphi - \\psi}$.\nThis bound gives both uniqueness and continuity, so the problem is well posed.\nNon-example: the Cauchy problem for the Laplace equation fails clause 3; see [[Hadamard's example|SZ6MEVM89M]].\nThe backward heat equation also fails clause 3, and on bounded domains it fails clause 1 for most data.\n\n## Explanations\n\n### L1 human\n\nA well-posed problem has exactly one solution for every datum, and the solution changes little when the datum changes little.\n",
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