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Global regularity of the \(\overline{\partial}\)-Neumann problem on circular domains. (English) Zbl 0662.32016
We show that if \(D\subseteq {\mathbb{C}}^ n\), \(n\geq 2\), is a smooth bounded pseudoconvex circular domain with \(\sum^{n}_{i=1}z_ i\frac{\partial r}{\partial z_ i}\neq 0\) near the boundary, where r(z) is the defining function for D, then global regularity of the \({\bar \partial}\)-Neumann problem holds on D. More precisely, if given \(f\in W^ k_{p,q}(D)\), let \(u\in L^ 2_{q,p}(D)\) be the solution to the \({\bar \partial}\)- Neumann problem, \[ Q(u,v)=({\bar \partial}u,{\bar \partial}v)+({\vec \partial}u,{\vec \partial}v)=(f,v), \] for all \(v\in \tilde {\mathcal D}_{p,q}(D)\), then \(u\in W^ k_{p,q}(D)\) and \(\| u\|_ k\leq C_ k\| f\|_ k\), where \(W^ k_{p,q}(D)\) is the Sobolev space of order k on D and \(\tilde {\mathcal D}_{p,q}(D)\) is the completion of smooth (p,q)-forms with Neumann boundary conditions under Q. We also prove that same conclusions hold on any smooth bounded Reinhardt pseudoconvex domain. As an application we show that same results hold on Sibony’s domain that is a smooth bounded weakly pseudoconvex domain in \({\mathbb{C}}^ 3\), but fails to have sup-norm estimate for the \({\bar \partial}\)-equation.
Reviewer: S.Chen

32W05 \(\overline\partial\) and \(\overline\partial\)-Neumann operators
32A07 Special domains in \({\mathbb C}^n\) (Reinhardt, Hartogs, circular, tube) (MSC2010)
32T99 Pseudoconvex domains
35N15 \(\overline\partial\)-Neumann problems and formal complexes in context of PDEs
Full Text: DOI EuDML
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