TY - JOUR
T1 - Hearing pseudoconvexity in Lipschitz domains with holes via ∂¯
AU - Fu, Siqi
AU - Laurent-Thiébaut, Christine
AU - Shaw, Mei Chi
N1 - Publisher Copyright:
© 2017, Springer-Verlag Berlin Heidelberg.
Copyright:
Copyright 2017 Elsevier B.V., All rights reserved.
PY - 2017/12/1
Y1 - 2017/12/1
N2 - 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.
AB - 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.
KW - Dolbeault cohomology
KW - L-Dolbeault cohomology
KW - Pseudoconvexity
KW - Serre duality
KW - ∂¯ -Neumann Laplacian
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U2 - 10.1007/s00209-017-1863-6
DO - 10.1007/s00209-017-1863-6
M3 - Article
AN - SCOPUS:85013170665
VL - 287
SP - 1157
EP - 1181
JO - Mathematische Zeitschrift
JF - Mathematische Zeitschrift
SN - 0025-5874
IS - 3-4
ER -