Approximated harmonic maps with tension fields in Zygmund class (Q6658225)

Assuming that \(u\) is a map from \(D_8\) to a compact smooth Riemannian manifold \(N\) with bounded energy, it is proven that there exists a constant \(\lambda>0\) which depends only on \(N\) and on \(E(u,D_8)\) such that if the tension field \(\tau\) belongs to the Zygmund class \(L\ln^{\lambda}L(D_8)\), then the Hopf differential of \(u\) belongs to the Zygmund class \(L\ln^{3}L(D_1)\) and the norm \(\|h\|_{L\ln^{3}L(D_1)}\) is only dependent on \(N\), \(E(u,D_8)\) and \(\|\tau\|_{L\ln^{\lambda}L(D_8)}\). This also implies the energy identity and the necklessness of a blow-up sequence \(u_n\) with bounded energy \(E(u_n)\) and bounded \(\tau(u_n)\) in \(L\ln^{\lambda}L(D_8)\).