Solutions of Cauchy-Riemann equations on pseudoconvex domain with nonsmooth boundary

Seongan Lim Yie, Purdue University

Abstract

We want to prove global regularity of the $\bar\partial$-Problem on pseudoconvex domains in $\doubc\sp{n}$ with $C\sp2$ boundary. First of all, we prove estimates of a solution for pseudoconvex domains with smooth boundaries by studying precisely the dependence of the regularity of the solution on the smoothness of the boundary. And then we prove that a pseudoconvex domain with $C\sp2$ boundary can be approximated nicely by pseudoconvex domains with smooth boundaries from the inside of the domain. Finally, we prove global regularity of the $\bar\partial$-Problem by showing that the solution loses about two-thirds of the derivatives. The solution we have for this regularity is obtained as the limit of a sequence of approximate solutions which we obtain by smoothing the extension of the solution of a $\bar\partial$-Problem in each approximating domain. Also we prove that if a pseudoconvex domain has $C\sp4$ boundary, then there is a solution with loss of only three derivatives.

Degree

Ph.D.

Advisors

Catlin, Purdue University.

Subject Area

Mathematics

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