Document Type

Article

Date

3-16-2009

Embargo Period

11-11-2011

Disciplines

Mathematics

Description/Abstract

As long ago as 1962 Nitsche conjectured that a harmonic homeomorphism h: A(r,R) onto-> A(r*, R*) between planar annuli exists if and only if R*/r* > 1/2 ((R/r) + (r/R)). We prove this conjecture when the domain annulus is not too wide; explicitly, when log(R/r) < 3/2. For general A(r,R) the conjecture is proved under additional assumption that either h or its normal derivative have vanishing average on the inner boundary circle. This is the case for the critical Nitsche mapping which yields equality in the above inequality. The Nitsche mapping represents so-called free evolution of circles of the annulus A(r,R). It will be shown on the other hand that forced harmonic evolution results in greater ratio R*/r*. To this end, we introduce the underlying differential operators for the circular means of the forced evolution and use them to obtain sharp lower bounds of R*/r*.

Additional Information

This manuscript is from arXiv.org, for more information see http://arxiv.org/abs/0903.2665

Source

Harvested from arXiv.org

Creative Commons License


This work is licensed under a Creative Commons Attribution 3.0 License.

Included in

Mathematics Commons

Share

COinS
 
 

To view the content in your browser, please download Adobe Reader or, alternately,
you may Download the file to your hard drive.

NOTE: The latest versions of Adobe Reader do not support viewing PDF files within Firefox on Mac OS and if you are using a modern (Intel) Mac, there is no official plugin for viewing PDF files within the browser window.