Resumen de Bubbling location for $F$-harmonic maps and inhomogeneous Landau--Lifshitz equations

Yuxiang Li, Youde Wang

• 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 $f$