Hearing pseudoconvexity in Lipschitz domains with holes via $\overline\partial$

Siqi Fu (1) , Christine Laurent-Thiébaut (2) , Mei-Chi Shaw (3)

1 Rutgers University [Camden]
2 IF - Institut Fourier
3 UND - University of Notre Dame [Indiana]

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.

Mots clés: L 2 -Dolbeault cohomology; Dolbeault cohomology; ∂-Neumann Laplacian; pseudoconvexity; Serre duality

Domaines: Variables complexes [math.CV]

Siqi Fu, Christine Laurent-Thiébaut, Mei-Chi Shaw. Hearing pseudoconvexity in Lipschitz domains with holes via $\overline\partial$. Mathematische Zeitschrift, 2017, 287 (3-45), pp.1157-1181. ⟨hal-01398638⟩