# Commentarii Mathematici Helvetici

Full-Text PDF (121 KB) | Metadata | Table of Contents | CMH summary

**Volume 81, Issue 2, 2006, pp. 433–448**

**DOI: 10.4171/CMH/57**

Published online: 2006-06-30

Bubbling location for *F*-harmonic maps and inhomogeneous Landau–Lifshitz equations

^{[1]}and Pigong Han

^{[2]}(1) ICTP, Trieste, Italy

(2) Chinese Academy of Sciences, Beijing, China

Let $f$ be a positive smooth function on a closed Riemann surface $(M,g)$. The $f$-energy of a map $u$ from $M$ to a Riemannian manifold $(N,h)$ is defined as $$E_f(u)=\int_Mf|\nabla u|^2 dV_g.$$ In this paper, we will study the blow-up properties of Palais--Smale sequences for $E_f$. We will show that, if a Palais--Smale sequence is not compact, then it must blow up at some critical points of $f$. As a consequence, if an inhomogeneous Landau--Lifshitz system, i.e. a solution of $$u_t=u\times\tau_f(u)+\tau_f(u), u: M\rightarrow S^2,$$ blows up at time $\infty$, then the blow-up points must be the critical points of $f$.

*Keywords: **f*-harmonic map, inhomogeneous Landau–Lifshitz equation, *f*-harmonic flow, blow-up point

Najib Salah, Han Pigong: Bubbling location for *F*-harmonic maps and inhomogeneous Landau–Lifshitz equations. *Comment. Math. Helv.* 81 (2006), 433-448. doi: 10.4171/CMH/57