Commentarii Mathematici Helvetici


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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

Salah Najib[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