Hearing pseudoconvexity in Lipschitz domains with holes via ∂¯

Let Ω = Ω ~ \ D¯ where Ω ~ is a bounded domain with connected complement in Cⁿ (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 of Ω on various function spaces. In particular, we show that if the boundaries of Ω ~ and D are Lipschitz and C²-smooth respectively, then both Ω ~ and D are pseudoconvex if and only if 0 is not in the spectrum of the ∂¯ -Neumann Laplacian of Ω on (0, q)-forms for 1 ≤ q≤ n- 2 when n≥ 3 ; or 0 is not a limit point of the spectrum of the ∂¯ -Neumann Laplacian on (0, 1)-forms when n= 2.