Liouville定理的另一种证法

Another Proof of Liouville Theorem

设Ω是Rn 上的一个开子集 ,如果Sobolev类映射 f∈W1,nloc(Ω ,Rn)是一拟正则映射 ;那么f或是一个常数 ,或是 Rn=Rn∪ {∞ }上M bius变换在Ω上的限制 .当维数n为偶数时 ,映射 f的正则性假定可降到 f∈W1,n/2loc (Ω ,Rn)

Let Ω be an open subset of R n and f:Ω→R n be a mapping of Sobolev classes W 1,n loc (Ω, R n). if f is a 1-quasi regular mapping, then f is either constant or the restriction of a Mbius transformation of R n on Ω. Particularly, if the dimension n is an even number (n=2l,l∈N), the regularity hyperthesis of f may be weaken to the sharp condition f∈W 1,n/2 loc (Ω,R n).