Bubbling analysis near the Dirichlet boundary for approximate harmonic maps from surfaces

Jürgen Jost, Lei Liu, and Miaomiao Zhu

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Submission date: 29. Jun. 2016
Pages: 23
published in: Communications in analysis and geometry, 27 (2019) 3, p. 639-669 
DOI number (of the published article): 10.4310/CAG.2019.v27.n3.a5
Bibtex
MSC-Numbers: 53C43, 58E20
Keywords and phrases: harmonic map, heat flow, Dirichlet boundary, blow-up, energy identity, no neck
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Abstract:
For a sequence of maps with a Dirichlet boundary condition from a compact Riemann surface with smooth boundary to a general compact Riemannian manifold, with uniformly bounded energy and with uniformly L²-bounded tension field, we show that the energy identity and the no neck property hold during a blow-up process near the Dirichlet boundary. We apply these results to the two dimensional harmonic map flow with Dirichlet boundary and prove the energy identity at finite and infinite singular time. Also, the no neck property holds at infinite time.