The qualitative behavior at the free boundary for approximate harmonic maps from surfaces
Let {u_n\{un} be a sequence of maps from a compact Riemann surface M with smooth boundary to a general compact Riemannian manifold N with free boundary on a smooth submanifold $K\subset N$$KN satisfying \begin{aligned} \sup _n \left( \Vert \nabla u_n\Vert _{L^2(M)\Vert \tau (u_n)\Vert _{L^2(M)\right) \le \Lambda \end{aligned}$supn||un||L2(M)|t(un)|L2(M)=L,where \tau (u_n)$t(un) is the tension field of the map $u_n$$un. We show that the energy identity and the no neck property hold during a blow up process. The assumptions are such that this result also applies to the harmonic map heat flow with free boundary, to prove the energy identity at finite singular time as well as at infinity time. Also, the no neck property holds at infinity time.
Mathematische annalen