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

TL;DR

This paper investigates the boundary behavior of approximate harmonic maps from surfaces, establishing energy identity and no neck properties during blow-up, with applications to harmonic map heat flow with free boundary.

Contribution

It proves energy identity and no neck property for approximate harmonic maps with free boundary, extending results to heat flow at finite and infinite times.

Findings

Energy identity holds during blow-up.

No neck property is established at infinity time.

Results apply to harmonic map heat flow with free boundary.

Abstract

Let $\{u_n\}$ 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 $K\subset N$ satisfying \[ \sup_n \ \left(\|\nabla u_n\|_{L^2(M)}+\|\tau(u_n)\|_{L^2(M)}\right)\leq \Lambda, \] where $\tau(u_n)$ is the tension field of the map $u_n$. We show that the energy identity and 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.