A general method of weights in the d-bar-Neumann problem
arXiv:1001.5093
Abstract
This thesis deals with Partial Differential Equations in Several Complex Variables and especially focuses on a general estimate for the -Neumann problem on a domain which is -pseudoconvex or -pseudoconcave at a boundary point . Generalizing Property () by \cite{C84}, we define Property $(f\T-\M\T-P)^k$ at . This property yields the estimate {(f\T-\M)^k} \qquad \no{f(Λ)\mathcal M u}^2\le c(\no{\bar\partial u}^2+\no{\bar\partial^*u}^2+\no{u}^2)+C_\M\no{u}^2_{-1} for any $u\in C^\infty_c(U\cap \barΩ)^k\cap \T{Dom}(\dib^*)$ where is a neighborhood of . We want to point out that under a suitable choice of and $\M$, $(f\T-\M)^k$ is the subelliptic, superlogarithmic, compactness and subelliptic multiplier estimate. The thesis also aims at exhibiting some relevant classes of domains which enjoy Property $(f\T-\M\T-P)^k$ and at discussing recent literature on the -Neumann problem in the framework of this property.
A Thesis submitted for the degree of Doctor of Philosophy in front of the Committee composed by Joseph J. Kohn (President); Jeffery D. Mc.Neal; Emil J. Straube. Supervisor: Giuseppe Zampieri. 121 pages