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We have decided to discontinue the publication of preprints on our preprint server as of 1 March 2024. The publication culture within mathematics has changed so much due to the rise of repositories such as ArXiV (www.arxiv.org) that we are encouraging all institute members to make their preprints available there. An institute's repository in its previous form is, therefore, unnecessary. The preprints published to date will remain available here, but we will not add any new preprints here.

MiS Preprint
26/2016

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

Jürgen Jost, Lei Liu and Miaomiao Zhu

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 \[\sup_n(\|\nabla u_n\|_{L^2(M)}+\|\tau(u_n)\|_{L^2(M)})\leq \Lambda,\]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.

Received:
21.03.16
Published:
21.03.16
Keywords:
harmonic map, heat flow, free boundary, blow up, energy identity, no neck

Related publications

inJournal
2019 Journal Open Access
Jürgen Jost, Lei Liu and Miaomiao Zhu

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

In: Mathematische Annalen, 374 (2019) 1-2, pp. 133-177