数学所周五学术报告—L^2-Dolbeault Cohomology Groups on Annuli
主 题: 数学所周五学术报告—L^2-Dolbeault Cohomology Groups on Annuli
报告人: Professor Mei-Chi Shaw (University of Notre Dame)
时 间: 2017-06-16 15:00-16:00
地 点: 理科一号楼1114
Abstract: The Dolbeault cohomology measures the obstruction to solving the Cauchy-Riemann equations. The range of the Cauchy-Riemann operator is closed in some topological space if and only if the corresponding Dolbeault cohomology group is Hausdorff.
In this talk we will report some recent new results on the L^2 closed range property for $\bar\partial$ on an annulus between two pseudoconvex domains, when the inner domain is not smooth. In particular, we show the Hausdorff property of the L^2 Dolbeault cohomology group on a domain between a ball and a bi-disc, the so-called Chinese Coin problem. Our methods also give Sobolev W^1-estimates for the $\bar\partial$-equations on non-smooth domains, including certain product domains or intersection of smooth bounded pseudoconvex domains. One can characterize Lipschitz domains with holes through their L^2 Dolbeault cohomology (joint work with Debraj Chakrabarti, Siqi Fu, and Christine Laurent-Thi\'ebaut).