Hearing pseudoconvexity in Lipschitz domains with holes via $\overline\partial$
Résumé
Let Ω = Ω \ D where Ω is a bounded domain with connected complement in C n (or more generally in a Stein manifold) and D is relatively compact open subset of Ω with connected complement in Ω. We obtain characterizations of pseudoconvexity of Ω and D through the vanishing or Hausdorff property of the Dolbeault cohomology groups on various function spaces. In particular, we show that if the boundaries of Ω and D are Lipschitz and C 2-smooth respectively, then both Ω and D are pseudoconvex if and only if 0 is not in the spectrum of the ∂-Neumann Laplacian on (0, q)-forms for 1 ≤ q ≤ n − 2 when n ≥ 3; or 0 is not a limit point of the spectrum of the ∂-Neumannn Laplacian on (0, 1)-forms when n = 2. Mathematics Subject Classification (2000): 32C35, 32C37, 32W05.
Origine | Fichiers produits par l'(les) auteur(s) |
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