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  "body": "\n## Problem\n\nLet $\\Omega \\subseteq \\R^n$ be a bounded open set.\nConsider a regular problem of the [[calculus of variations|PQ25NHV0KS]]: minimise $J(u) = \\int_\\Omega L(x, u, \\nabla u) \\, dx$ with given values of $u$ on $\\partial\\Omega$.\nHere the Lagrangian $L$ is smooth and strictly convex in $\\nabla u$.\nHilbert asked whether every such problem has a solution.\nHe allowed suitable assumptions on the [[boundary conditions|TM8VQJWGHZ]] and, if needed, a suitable extension of the notion of solution.\n\n## Remarks\n\nThe problem is a programme rather than a single statement, and its answer depends on how \"suitable\" is made precise.\nThe modern answer is largely affirmative.\nThe [[direct method|TE379HBTY0]] gives minimisers in Sobolev spaces for Lagrangians that are convex in $\\nabla u$ and satisfy growth conditions, which is the \"suitably extended\" notion of solution.\nRegularity theory, which grew from [[Hilbert's nineteenth problem|QPMM2ZPWBW]], then shows that these generalised solutions are classical in many cases.\nThe same circle of ideas gives weak solutions of boundary value problems for linear and nonlinear elliptic equations.\n\n## Explanations\n\n### L1 human\n\nHilbert's twentieth problem asks whether every reasonable minimisation problem with boundary values has a solution, if one is willing to widen what counts as a solution.\nThe twentieth century answered this mostly in the affirmative, by working with weak solutions in Sobolev spaces and then proving that they are in fact smooth.\n",
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      "first_stated": "1900",
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