Hearing pseudoconvexity in Lipschitz domains with holes via \({\bar{\partial }}\)

  • Siqi Fu
  • Christine Laurent-Thiébaut
  • Mei-Chi Shaw
Article

Abstract

Let \(\Omega ={\widetilde{\Omega }}{\setminus } \overline{D}\) where \({\widetilde{\Omega }}\) is a bounded domain with connected complement in \({\mathbb {C}}^n\) (or more generally in a Stein manifold) and D is relatively compact open subset of \({\widetilde{\Omega }}\) with connected complement in \(\widetilde{\Omega }\). We obtain characterizations of pseudoconvexity of \({\widetilde{\Omega }}\) and D through the vanishing or Hausdorff property of the Dolbeault cohomology groups of \(\Omega \) on various function spaces. In particular, we show that if the boundaries of \({\widetilde{\Omega }}\) and D are Lipschitz and \(C^2\)-smooth respectively, then both \({\widetilde{\Omega }}\) and D are pseudoconvex if and only if 0 is not in the spectrum of the \(\overline{\partial }\)-Neumann Laplacian of \(\Omega \) on (0, q)-forms for \(1\le q\le n-2\) when \(n\ge 3\); or 0 is not a limit point of the spectrum of the \(\overline{\partial }\)-Neumann Laplacian on (0, 1)-forms when \(n=2\).

Keywords

Dolbeault cohomology \(L^2\)-Dolbeault cohomology Serre duality \(\overline{\partial }\)-Neumann Laplacian Pseudoconvexity 

Mathematics Subject Classification

32C35 32C37 32W05 

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Copyright information

© Springer-Verlag Berlin Heidelberg 2017

Authors and Affiliations

  • Siqi Fu
    • 1
  • Christine Laurent-Thiébaut
    • 2
  • Mei-Chi Shaw
    • 3
  1. 1.Department of Mathematical SciencesRutgers UniversityCamdenUSA
  2. 2.Institut FourierUniversité Grenoble-Alpes, Centre National de la Recherche Scientifique (CNRS)GrenobleFrance
  3. 3.Department of MathematicsUniversity of Notre DameNotre DameUSA

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