The unique extremality counterexample
The unique extremality counterexample
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DOI：
10.1007/bf02788705
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发表时间：
1998-12
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通讯作者：
E. Reich
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文献类型：
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作者：
E. Reich
假设f是单位磁盘a ={Izl< x)到自身的拟共形映射。令# f (z)= fe/f~表示A中的复膨胀ae，令k [f]= esssup {l# s (z) l: zc A}= II~ sll~。如果假设(i) g是A的拟共形映射，且其在0A上的点向边值与点向边值一致，且（ii） k [g]< k [f]暗示g= f，则映射f在A的拟共形映射集合中是唯一极值的。通过简单的变分考虑，很容易证明如果f是唯一极值的，则
Suppose f is a quasiconformal mapping of the unit disk A={Izl< x) onto itself. Let# f (z)= fe/f~ denote the complex dilatation ae in A, and put k [f]= esssup {l# s (z) l: zc A}= II~ sll~.The mapping f is uniquely extremal among the collection of quasiconformal mappings of A if the assumptions (i) g is a quasiconformal mapping of A whose pointwise boundary values on 0A agree with those off, and (ii) k [g]< k [f], imply that g= f. By means of simple variational considerations it is very easy to show that if f is uniquely extremal then