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

TL;DR

This work studies bubbling phenomena for maps u: M -> N that minimize or are critical for the inhomogeneous energy E_f(u)=∫_M f|∇u|^2 dV_g on a closed Riemann surface, showing that any noncompact Palais-Smale sequence must blow up only at critical points of f; similarly, for the inhomogeneous Landau-Lifshitz flow u_t = u × τ_f(u) + τ_f(u) on S^2, blow-up at time infinity occurs at such points. The authors develop a generalized removable singularity theory inspired by Sacks-Uhlenbeck, adapt the Hopf differential analysis to the f-weighted setting, and derive a variational framework for E_f, including a domain-variation formula. Their main results include Theorem 1 (local extension/removability for punctured discs with τ(u)=α∇u+g), Theorem 2 (any blow-up point of solutions to fτ(u)+∇f·∇u=α_k with bounded energy is a critical point of f), and Theorem 3 (long-time blow-up points for the inhomogeneous Landau-Lifshitz equation occur at critical points of f). Consequently, the bubbling set is finite, the weak limit is an f-harmonic map away from bubbling points, and the results link the geometry of f to the dynamical behavior, informing the asymptotics of the inhomogeneous Landau-Lifshitz flow.

Abstract

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 $f$.