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  "body": "\n## Problem\n\nLet $\\Omega \\subseteq \\C^n$ be a bounded open set with smooth boundary, and let $1 \\le q \\le n$.\nLet $L^2_{(0,q)}(\\Omega)$ be the Hilbert space of [[(0,q)-forms|DC81TCR98Y]] on $\\Omega$ with square integrable coefficients.\nRegard $\\bar\\partial$ as a closed, densely defined operator from $L^2_{(0,q)}(\\Omega)$ to $L^2_{(0,q+1)}(\\Omega)$.\nIt acts in the sense of distributions on the forms for which the result is square integrable.\nLet $\\bar\\partial^*$ be its [[Hilbert space adjoint|86B7HF79PQ]], and let\n$$\\square = \\bar\\partial \\bar\\partial^* + \\bar\\partial^* \\bar\\partial$$\non the set of $u \\in L^2_{(0,q)}(\\Omega)$ with $u \\in \\operatorname{Dom}(\\bar\\partial) \\cap \\operatorname{Dom}(\\bar\\partial^*)$, $\\bar\\partial u \\in \\operatorname{Dom}(\\bar\\partial^*)$ and $\\bar\\partial^* u \\in \\operatorname{Dom}(\\bar\\partial)$.\nGiven $\\alpha \\in L^2_{(0,q)}(\\Omega)$, find $u$ in this domain with $\\square u = \\alpha$, and study the regularity of $u$ up to the boundary.\n\n## Remarks\n\nFor forms smooth up to the boundary, membership in the domain of $\\square$ means two boundary conditions, the $\\bar\\partial$-Neumann conditions.\nIf $\\rho$ is a defining function of $\\Omega$, they say that the contraction of $u$ with $\\bar\\partial \\rho$ and the contraction of $\\bar\\partial u$ with $\\bar\\partial \\rho$ both vanish on $\\partial\\Omega$.\nThese conditions are not elliptic, so the standard theory of elliptic boundary value problems does not apply.\nSpencer proposed the problem in the 1950s as a way to extend Hodge theory to manifolds with boundary.\nKohn solved it for strongly pseudoconvex domains in 1963: $\\square$ has a bounded inverse $N$, the Neumann operator.\nMoreover $N$ maps forms that are smooth up to the boundary to forms that are smooth up to the boundary.\nFor a bounded [[pseudoconvex|AHYZMSBBNH]] domain, Hörmander's $L^2$ estimates show that $\\square$ is invertible on $L^2_{(0,q)}(\\Omega)$ for $q \\ge 1$.\nWhen $\\bar\\partial \\alpha = 0$, the form $\\bar\\partial^* N \\alpha$ solves $\\bar\\partial v = \\alpha$ and is the solution of least $L^2$ norm.\nRegularity up to the boundary may fail for smooth bounded pseudoconvex domains: Christ showed that $N$ does not preserve $C^\\infty(\\overline{\\Omega})$ on worm domains.\n\n## Explanations\n\n### L1 human\n\nThe problem asks to invert a Laplace-type operator built from $\\bar\\partial$, with boundary conditions that come from the adjoint of $\\bar\\partial$ in a Hilbert space.\nOnce it is inverted, one gets a canonical solution of the equation $\\bar\\partial v = \\alpha$ with good estimates.\nThe difficulty is that the boundary conditions are weaker than in classical problems, so the behaviour at the boundary depends on the shape of the domain.\n",
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