Remarks on approximate harmonic maps in dimension two

Abstract

For the class of approximate harmonic maps $u\in W^{1,2}(\Sigma ,N)$ from a closed Riemmanian surface $(\Sigma ,g)$ to a compact Riemannian manifold $(N,h)$, we show that (i) the so-called energy identity holds for weakly convergent approximate harmonic maps $\{u_n\}:\Sigma \to N$, with tension fields $\tau (u_n)$ bounded in the Morrey space $M^{1,\delta }(\Sigma )$ for some $0\le \delta <2$; and (ii) if an approximate harmonic map $u$ has tension field $\tau (u)\in L\log L(\Sigma )\cap M^{1,\delta }(\Sigma )$ for some $0\le \delta <2$, then $u\in W^{2,1}(\Sigma ,N)$. Based on these estimates, we further establish the bubble tree convergence, referring to energy identity both $L^{2,1}$ of gradients and $L^1$-norm of hessians and the oscillation convergence, for a weakly convergent sequence of approximate harmonic maps $\{u_n\}$, with tension fields $\tau (u_n)$ uniformly bounded in $M^{1,\delta }(\Sigma )$ for some $0\le \delta <2$ and uniformly integrable in $L\log L(\Sigma )$.

25 pages

References in corpus (1)

• Energy identity for approximate harmonic maps from surface to general targets

Cited by in corpus (1)

• Boundary bubbling analysis of approximate harmonic maps under either weak or strong anchoring conditions in dimension two