#### 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_{*}.

#### Recommended Citation

Iwaniec, Tadeusz; Kovalev, Leonid V.; and Onninen, Jani, "Harmonic Mappings of an Annulus, Nitsche Conjecture and its Generalizations" (2009). *Mathematics Faculty Scholarship*. 60.

https://surface.syr.edu/mat/60

#### Source

Harvested from arXiv.org

#### Creative Commons License

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

## Additional Information

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