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Let $\{u_n\}$ be a sequence of maps from a compact Riemann surface $M$ withsmooth boundary to a general compact Riemannian manifold $N$ with free boundaryon a smooth submanifold $K\subset N$ satisfying \[ \sup_n \ \left(\|\nablau_n\|_{L^2(M)}+\|\tau(u_n)\|_{L^2(M)}\right)\leq \Lambda, \] where $\tau(u_n)$is the tension field of the map $u_n$. We show that the energy identity and theno neck property hold during a blow-up process. The assumptions are such thatthis result also applies to the harmonic map heat flow with free boundary, toprove the energy identity at finite singular time as well as at infinity time.Also, the no neck property holds at infinity time.

The qualitative behavior at the free boundary for approximate harmonic maps from surfaces