Boundary Behavior of Harmonic and Quasiregular Mappings

Boundary Behavior of Harmonic and Quasiregular Mappings
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  BOUNDARY BEHAVIOR OF HARMONICAND QUASIREGULAR MAPPINGS Antti RasilaDepartment of Mathematics and Systems AnalysisHelsinki University of Technology (TKK) antti.rasila@tkk.fi The 12th Romanian–Finnish SeminarAugust 17–21, 2009, Turku, Finland(Joint research with S. Ponnusamy)  Abstract We study the connection between multiplicities of the zeros andboundary behavior of bounded harmonic and quasiregularmappings of the plane.Sufficient conditions for the existence of angular (non-tangential)limit at a boundary point, provided that multiplicities of zeroes of the function grow fast enough on a given sequence of pointsapproaching the boundary. We compare these results and alsoconsider sharpness of such conditions in the planar case.This talk is based on joint research with S. Ponnusamy.  Introduction Lindel¨of’s theoremSuppose that  γ   is a curve, with parameter interval [0 , 1], such that | γ  ( t  ) | <  1 if   t   <  1 and  γ  (1) = 1. If   f    is a bounded analyticfunction of the unit disk  D  andlim t  → 1 f    γ  ( t  )  =  α, then  f    has angular limit  α  at 1, i.e. limit in each angular regioncontained in the unit disk with the vertex in 1.  Questions and remarks What about generalizations of Lindel¨of’s theorem for otherclasses of functions? Holds for quasiconformal mappings in any dimension  n ≥ 2.Does  not  hold for non-univalent quasiregular mappings for n ≥ 3 (Rickman).Lindel¨of type results have been proved by Vuorinen withvarious assumptions for quasiregular mappings in  R n . Terminology: H n is the upper half space and  B n is the unit ball.For a discrete and open mapping  f    :  G   → R n , the local(topological) index  i  ( x  , f    ) is the infimum of sup y   card f   − 1 ( y  ) ∩ U   where  U   runs through the neighborhoodsof   x  .  QR mappings ( n ≥ 2) For example, we may prove the following:Theorem (R., 2005)Let  f    :  H n → B n be a bounded  K  -quasiregular mapping, let t  k   = 2 − k  and  f    ( t  k  e  n ) = 0 for all  k   = 0 , 1 ,... (1) If limsup k  →∞ t  γ  k  µ ( t  k  e  n , f    ) = ∞ , where µ ( t  k  e  n , f    ) =  C  1 i  ( t  k  e  n , f    ) 1 / ( n − 1) and  γ   =  C  2  log(1 /β  ), then f    ≡ 0.(2) If   µ ( t  k  e  n , f    ) →∞  as  k   →∞ , then  f    has an angular limit 0 atthe srcin.What about the case  n  = 2?
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