Approximated harmonic maps with tension fields in Zygmund class

Abstract

Suppose that u is a map from $D_8$ to a compact smooth Riemannian manifold N with bounded energy. We show that there exists a constant $\lambda >0$ which depends only on N and $E(u,D_8)$ such that if the tension field τ belongs to Zygmund class $L{\ln }^{\lambda }L\left(D_8\right)$, then the Hopf differential of u belongs to the Zygmund class $L{\ln }^{3}L\left(D_1\right)$ and the norm ${\Vert h\Vert }_{L{\ln }^{3}L\left(D_1\right)}$ depends only on $N,E(u,D_8)$ and ${\Vert \tau \Vert }_{L{\ln }^{\lambda }L\left(D_8\right)}$. As a direct corollary, we obtain the energy identity and necklessness of a blow-up sequence $u_n$ with bounded energy $E\left(u_n\right)$ and bounded $\tau \left(u_n\right)$ in $L{\ln }^{\lambda }L\left(D_8\right)$.