View in EDS

Semi-global analyticity for the \(\overline{\partial}\)-Neumann problem in pseudo-convex domains

Saved in:
Bibliographic Details
Title: Semi-global analyticity for the \(\overline{\partial}\)-Neumann problem in pseudo-convex domains
Authors: Ben Moussa, Benoît
Publisher Information: Scuola Normale Superiore, Pisa
Subject Terms: \(\overline\partial\) Neumann problem, Pseudoconvex domains, Analytical consequences of geometric convexity (vanishing theorems, etc.), \(\overline\partial\) and \(\overline\partial\)-Neumann operators, pseudoconvex domains
Description: In this paper some results of the following type are proved: Let \(\Omega \) be a bounded, pseudoconvex domain in \(\mathbb C^n \) of class \(\mathcal C^{\infty},\) let \(D\) be a connected component of the set of degenericity \(\Delta \) of the Levi form on \(\Omega \) and let \(V(D)\) be a bounded, open neighborhood of \(D\) in \(\mathbb C^n\) which separates \(D\) from the other connected components of \(\Delta .\) Then for \(q \in \{1, \dots , n-1 \}\) and some further geometric conditions the solution to the \(\overline \partial \)-Neumann problem \(\square u=f \) for \(f\in L_q^2(\Omega) \cap \mathcal C_q^{\omega } (\overline \Omega \cap V(D))\) lies in \(\mathcal C_q^{\omega } (\overline \Omega \cap V(D)).\) The author also indicates that the further geometric conditions are mostly satisfied in \(\mathbb C^2\) and gives some interesting examples demonstrating differences to earlier results in this direction due to M. Derridj.
Document Type: Article
File Description: application/xml
Access URL: https://zbmath.org/1467772
Accession Number: edsair.c2b0b933574d..3d94268086c268da4b734ed5d82ba234
Database: OpenAIRE
Description
Abstract:In this paper some results of the following type are proved: Let \(\Omega \) be a bounded, pseudoconvex domain in \(\mathbb C^n \) of class \(\mathcal C^{\infty},\) let \(D\) be a connected component of the set of degenericity \(\Delta \) of the Levi form on \(\Omega \) and let \(V(D)\) be a bounded, open neighborhood of \(D\) in \(\mathbb C^n\) which separates \(D\) from the other connected components of \(\Delta .\) Then for \(q \in \{1, \dots , n-1 \}\) and some further geometric conditions the solution to the \(\overline \partial \)-Neumann problem \(\square u=f \) for \(f\in L_q^2(\Omega) \cap \mathcal C_q^{\omega } (\overline \Omega \cap V(D))\) lies in \(\mathcal C_q^{\omega } (\overline \Omega \cap V(D)).\) The author also indicates that the further geometric conditions are mostly satisfied in \(\mathbb C^2\) and gives some interesting examples demonstrating differences to earlier results in this direction due to M. Derridj.