TY - JOUR

T1 - Harmonic mappings of an annulus,nitsche conjecture and its generalizations

AU - Iwaniec, Tadeusz

AU - Kovalev, Leonid V.

AU - Onninen, Jani

PY - 2010/10

Y1 - 2010/10

N2 - 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 R ≤ e3/2r. We also treat the general annuli A(r, R), 0 < r < R < 8, and obtain the sharp Nitsche bound under additional assumption that either h or its normal derivative have vanishing average along the inner circle of A(r, R). We consider the family of Jordan curves in A(r*, R*) obtained as images under h of concentric circles in A(r, R). We refer to such family of Jordan curves as harmonic evolution of the inner boundary of A(r, R). In the borderline case R*r*= 1/2(R/r+ r/R)the evolution begins with zero speed. It will be shown, as a generalization of the Nitsche Conjecture, that harmonic evolution with positive initial speed results in greater ratio R*r*in the deformed (target) annulus. To every initial isnpeoeudr tgheenreercaolirzraetsipoonnodfs tahne uNnidtsecrlhyeinCgodnijfefcetruernet.ial operator which yields sharp lower bounds of R* in our generalization of the Nitsche Conjecture.

AB - 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 R ≤ e3/2r. We also treat the general annuli A(r, R), 0 < r < R < 8, and obtain the sharp Nitsche bound under additional assumption that either h or its normal derivative have vanishing average along the inner circle of A(r, R). We consider the family of Jordan curves in A(r*, R*) obtained as images under h of concentric circles in A(r, R). We refer to such family of Jordan curves as harmonic evolution of the inner boundary of A(r, R). In the borderline case R*r*= 1/2(R/r+ r/R)the evolution begins with zero speed. It will be shown, as a generalization of the Nitsche Conjecture, that harmonic evolution with positive initial speed results in greater ratio R*r*in the deformed (target) annulus. To every initial isnpeoeudr tgheenreercaolirzraetsipoonnodfs tahne uNnidtsecrlhyeinCgodnijfefcetruernet.ial operator which yields sharp lower bounds of R* in our generalization of the Nitsche Conjecture.

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U2 - 10.1353/ajm.2010.0000

DO - 10.1353/ajm.2010.0000

M3 - Article

AN - SCOPUS:77958612323

SN - 0002-9327

VL - 132

SP - 1397

EP - 1428

JO - American Journal of Mathematics

JF - American Journal of Mathematics

IS - 5

ER -