Communications in Analysis and Geometry

Volume 27 (2019)

Number 3

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

Pages: 639 – 669

DOI: https://dx.doi.org/10.4310/CAG.2019.v27.n3.a5

Authors

Jürgen Jost (Max Planck Institute for Mathematics in the Sciences, Leipzig, Germany; and Department of Mathematics, Leipzig University, Leipzig, Germany)

Lei Liu (Department of Mathematics, Tsinghua University, Beijing, China; and Max Planck Institute for Mathematics in the Sciences, Leipzig, Germany)

Miaomiao Zhu (School of Mathematical Sciences, Shanghai Jiao Tong University, Shanghai, China)

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 $L^2$-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.

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The research leading to these results has received funding from the European Research Council under the European Union’s Seventh Framework Program (FP7/2007-2013) / ERC grant agreement no. 267087. Miaomiao Zhu was supported in part by National Natural Science Foundation of China (No. 11601325).

Received 21 June 2016

Accepted 11 April 2017

Published 3 September 2019