Regularity of the \overline∂-Neumann problem

Chen, So-Chin

doi:10.1090/S0002-9939-1991-1049842-7

Regularity of the $\overline \partial$-Neumann problem
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by So-Chin Chen

Proc. Amer. Math. Soc. 111 (1991), 779-785

DOI: https://doi.org/10.1090/S0002-9939-1991-1049842-7

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Abstract:

In this paper we prove that locally there is no obstruction to global regularity for the $\bar \partial$-Neumann problem. By this we mean the following: Let $D$ be a smoothly bounded pseudoconvex domain in ${{\mathbf {C}}^n},n \geq 2$, and let $p \in {\mathbf {D}}$. Given any $m \in {{\mathbf {Z}}^ + }$, one can construct a smoothly bounded pseudoconvex subdomain $D_m \subseteq D$ such that $b D_m \cap bD$ contains an open neighborhood of $p$ in $bD$ and the $\bar \partial$-Neumann problem on ${D_m}$ is globally regular up to order $m$ in the sense of the Sobolev norm.

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