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

### Salah Najib

ICTP, Trieste, Italy### Pigong Han

Chinese Academy of Sciences, Beijing, China

## Abstract

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

$E_{f}(u)=∫_{M}f∣∇u∣_{2}dV_{g}.$

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

$u_{t}=u×τ_{f}(u)+τ_{f}(u),u:M→S_{2},$

blows up at time $∞$, then the blow-up points must be the critical points of $f$.

## Cite this article

Salah Najib, Pigong Han, Bubbling location for $F$-harmonic maps and inhomogeneous Landau–Lifshitz equations. Comment. Math. Helv. 81 (2006), no. 2, pp. 433–448

DOI 10.4171/CMH/57