Bubbling location for $F$-harmonic maps and Inhomogeneous Landau-Lifshitz equations

Details Bubbling location for $F$-harmonic maps and Inhomogeneous Landau-Lifshitz equations Bubbling location for $F$-harmonic maps and Inhomogeneous Landau-Lifshitz equations

2005-04-25

arxiv.org/abs/math/0504502v2

Mathematics

Analysis of PDEs

13pages

Scientific paper

Let $f$ be a positive smooth function on a close Riemann surface (M,g). The $f-energy$ of a map $u$ from $M$ to a Riemannian manifold $(N,h)$ is defined as $$E_f(u)=\int_Mf|\nabla u|^2dV_g.$$ 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 blows up at some critical points of $f$. As a sequence, if an inhomogeneous Landau-Lifshitz system, i.e. a solution of $$u_t=u\times\tau_f(u)+\tau_f(u),\s u:M\to S^2$$ blows up at time $\infty$, then the blow-up points must be the critical points of $f$.

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