Liouville theorems for quasi-harmonic functions

✍ Scribed by Xiangrong Zhu; Meng Wang

• Publisher: Elsevier Science
• Year: 2010
• Tongue: English
• Weight: 261 KB
• Volume: 73
• Category: Article
• ISSN: 0362-546X
• DOI: 10.1016/j.na.2010.06.045

✦ Synopsis

Let N be a compact Riemannian manifold. A self-similar solution for the heat flow is a harmonic map from (R n , e -|x| 2 /2(n-2) ds 2 0 ) to N (n ≥ 3), which was also called a quasiharmonic sphere (cf. Lin and Wang (1999) [1]). (Here ds 2 0 is the Euclidean metric in R n .)

It arises from the blow-up analysis of the heat flow at a singular point. When N = R and without the energy constraint, we call this a quasi-harmonic function. In this paper, we prove that there is neither a nonconstant positive quasi-harmonic function nor a nonconstant L p (R n , e -|x| 2 /2(n-2) ds 2 0 )(p > n n-2 ) quasi-harmonic function. However, for all 1 ≤ p ≤ n/(n -2), there exists a nonconstant quasi-harmonic function in L p (R n , e -|x| 2 /2(n-2) ds 2 0 ).