Subelliptic estimates for the \({\bar \partial}\)-Neumann problem on pseudoconvex domains. (English) Zbl 0627.32013

Catlin, David

Ann. Math. (2) 126, 131-191 (1987).

This article gives a necessary and sufficient condition for the \({\bar \partial}\)-Neumann problem on a bounded pseudoconvex domain \(\Omega\) in \({\mathbb{C}}^ n\) with smooth boundary \(b\Omega =\{r=0\}\) to satisfy local subelliptic estimate at a boundary point \(z_ 0\in b\Omega\). The condition is given by \(D_ q(z_ 0)<\infty\), where q is the degree of the antiholomorphic part of the space of forms under consideration and \(D_ q(z_ 0)=\sup_{V^ q} \{\tau (V^ q,z_ 0)\}.\) Here the supremum is taken over germs \(V^ q\) of q-dimensional subvarieties passing through \(z_ 0\) and lying outside \(\Omega\), \(\tau (V^ q,z_ 0)\) denotes the maximum value of \(ord_ 0(r\circ \gamma^ k_ S)/ord_ 0(\gamma^ k_ S)\) among the one-dimensional components \(V^ q_{S,k}=\{z=\gamma^ k_ S(t):({\mathbb{C}},0)\to ({\mathbb{C}}^ n,z_ 0)\}\), \(k=1,...,P\) of the intersection \(V^ q\cap S\) by an \((n-q+1)-\) dimensional plane S in general position. This result gives a final answer to the problem initiated and studied by J. J. Kohn.

Reviewer: A.Kaneko

MSC:

32W05	\(\overline\partial\) and \(\overline\partial\)-Neumann operators
32T99	Pseudoconvex domains
35N15	\(\overline\partial\)-Neumann problems and formal complexes in context of PDEs
65H10	Numerical computation of solutions to systems of equations
35J55	Systems of elliptic equations, boundary value problems (MSC2000)

Keywords:

\({\bar \partial }\)-Neumann problem; pseudoconvex domain; subelliptic estimate