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Bubbling analysis near the Dirichlet boundary for approximate harmonic maps from surfaces
Jürgen Jost, Lei Liu and Miaomiao Zhu
For a sequence of maps with a Dirichlet boundary condition from a compact Riemann surface with smooth boundary to a general compact Riemannian manifold, with uniformly bounded energy and with uniformly L2-bounded tension field, we show that the energy identity and the no neck property hold during a blow-up process near the Dirichlet boundary. We apply these results to the two dimensional harmonic map ﬂow with Dirichlet boundary and prove the energy identity at finite and infinite singular time. Also, the no neck property holds at infinite time.