JournalscmhVol. 81 , No. 2DOI 10.4171/cmh/57

Bubbling location for <em>F</em>-harmonic maps and inhomogeneous Landau–Lifshitz equations

  • Salah Najib

    ICTP, Trieste, Italy
  • Pigong Han

    Chinese Academy of Sciences, Beijing, China
Bubbling location for <em>F</em>-harmonic maps and inhomogeneous Landau–Lifshitz equations cover

Abstract

Let ff be a positive smooth function on a closed Riemann surface (M,g)(M,g). The ff-energy of a map uu from MM to a Riemannian manifold (N,h)(N,h) is defined as

Ef(u)=Mfu2dVg.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 EfE_f. We will show that, if a Palais--Smale sequence is not compact, then it must blow up at some critical points of ff. As a consequence, if an inhomogeneous Landau--Lifshitz system, i.e. a solution of

ut=u×τf(u)+τf(u),u:MS2,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 ff.