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Type of Document Dissertation Author Harrington, Phillip S Author's Email Address harrington.19@nd.edu URN etd-04132004-090527 Title Compactness and Subellipticity for the D-Bar Neumann Operator on Domains with Minimal Smoothness Degree Doctor of Philosophy Department Mathematics Advisory Committee
Advisor Name Title Ikaros Bigi Committee Chair Alex Himonas Committee Member Jianguo Cao Committee Member Mei-Chi Shaw Committee Member Nancy Stanton Committee Member Keywords
- non-coercive boundary value problems
- partial differential equations
- several complex variables
Date of Defense 2004-03-30 Availability unrestricted Abstract In this thesis, we shall examine a strong form of Oka's Lemma which provides sufficient conditions for compact and subelliptic estimates for the d-bar Neumann operator on Lipschitz domains. On smooth domains, the condition for subellipticity is equivalent to D'Angelo finite-type and the condition for compactness is equivalent to Catlin's condition (P).
Once the basic properties of this condition have been established, we will study the extent to which these estimates can be extended to higher order derivatives on C^k domains, with k greater than or equal to 2. For the Lipschitz case, we will look at higher order estimates in the special case when the domain admits a plurisubharmonic defining function.
Finally, we will use these estimates to construct a compact solution operator for the boundary complex.
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