Preprint 26/2016
The qualitative behavior at the free boundary for approximate harmonic maps from surfaces
Jürgen Jost, Lei Liu, and Miaomiao Zhu
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Submission date: 21. Mar. 2016 (revised version: September 2018)
Pages: 39
published in: Mathematische Annalen, 374 (2019) 1-2, p. 133-177 
DOI number (of the published article): 10.1007/s00208-018-1759-8
Bibtex
Keywords and phrases: harmonic map, heat flow, free boundary, blow up, energy identity, no neck
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Abstract:
Abstract
Let {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 ⊂ N satisfying
<center class="math-display"> <img src="/fileadmin/preprint_img/2016/tex_2165a0x.png" alt="sup(∥∇un ∥L2(M) +∥τ(un)∥L2(M))≤ Λ, n " class="math-display"></center>where τ(un) is the tension field of the map 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.