A Remark on Irregularity of the ∂-neumann Problem on Non-smooth Domains
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چکیده
It is an observation due to J.J. Kohn that for a smooth bounded pseudoconvex domain Ω in C there exists s > 0 such that the ∂-Neumann operator on Ω maps W s (0,1)(Ω) (the space of (0, 1)-forms with coefficient functions in L -Sobolev space of order s) into itself continuously. We show that this conclusion does not hold without the smoothness assumption by constructing a bounded pseudoconvex domain Ω in C, smooth except at one point, whose ∂-Neumann operator is not bounded on W s (0,1)(Ω) for any s > 0. Let W (Ω) and W s (p,q)(Ω) denote the L -Sobolev space on Ω of order s and the space of (p, q)-forms with coefficient functions in W (Ω), respectively. Also ‖.‖s,Ω denotes the norms onW s (p,q)(Ω). Let Nq denote the inverse of the complex Laplacian, ∂∂ ∗ + ∂ ∗ ∂, on square integrable (0, q)-forms. It is an observation of Kohn, as the following proposition says, that on a smooth bounded pseudoconvex domain the ∂-Neumann problem is regular in the Sobolev scale for sufficiently small levels. Proposition 1 (Kohn). Let Ω be a smooth bounded pseudoconvex domain in C. There exist positive ε and C (depending on Ω) such that ‖Nqu‖ε,Ω ≤ C‖u‖ε,Ω, ‖∂Nqu‖ε,Ω ≤ C‖u‖ε,Ω, ‖∂ ∗ Nqu‖ε.Ω ≤ C‖u‖ε,Ω for u ∈W s (0,q)(Ω) and 1 ≤ q ≤ n. We show that if one drops the smoothness assumption then the ∂-Neumann operator, N1, may not map any positive Sobolev space into itself continuously. Theorem 1. There exists a bounded pseudoconvex domain Ω in C, smooth except one point, such that the ∂-Neumann operator on Ω is not bounded on W s (0,1)(Ω) for any s > 0. Proof. We will build the domain by attaching infinitely many worm domains (constructed by Diederich and Fornæss in [DF77]) with progressively larger winding. Let Ωj be a worm domain, a smooth bounded pseudoconvex domain, in C 2 that winds 2πj such that Ωj ⊂ {(z, w) ∈ C 2 : |z| < 2, 4 < |w| < 42} Date: March 11, 2008. 2000 Mathematics Subject Classification. 32W05.
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تاریخ انتشار 2006