dbar-Neumann problem

Remarks

For forms smooth up to the boundary, membership in the domain of $\square$ means two boundary conditions, the $\bar\partial$-Neumann conditions. If $\rho$ 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 $\partial\Omega$. These conditions are not elliptic, so the standard theory of elliptic boundary value problems does not apply. Spencer proposed the problem in the 1950s as a way to extend Hodge theory to manifolds with boundary. Kohn solved it for strongly pseudoconvex domains in 1963: $\square$ has a bounded inverse $N$, the Neumann operator. Moreover $N$ maps forms that are smooth up to the boundary to forms that are smooth up to the boundary. For a bounded pseudoconvex domain, Hörmander's $L^2$ estimates show that $\square$ is invertible on $L^2_{(0,q)}(\Omega)$ for $q \ge 1$. When $\bar\partial \alpha = 0$, the form $\bar\partial^* N \alpha$ solves $\bar\partial v = \alpha$ and is the solution of least $L^2$ norm. Regularity 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.