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Transactions of the American Mathematical Society
Transactions of the American Mathematical Society
ISSN 1088-6850(online) ISSN 0002-9947(print)

 

Harmonic map heat flow with rough boundary data


Author: Lu Wang
Journal: Trans. Amer. Math. Soc. 364 (2012), 5265-5283
MSC (2010): Primary 58E20; Secondary 35K55
Published electronically: May 29, 2012
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Abstract: Let $ B_1$ be the unit open disk in $ \mathbb{R}^2$ and $ M$ a closed Riemannian manifold. In this note, we first prove the uniqueness for weak solutions of the harmonic map heat flow in $ H^1([0,T]\times B_1,M)$ whose energy is non-increasing in time, given initial data $ u_0\in H^1(B_1,M)$ and boundary data $ \gamma =u_0\vert _{\partial B_1}$. Previously, this uniqueness result was obtained by Rivière (when $ M$ is the round sphere and the energy of initial data is small) and Freire (when $ M$ is an arbitrary closed Riemannian manifold), given that $ u_0\in H^1(B_1,M)$ and $ \gamma =u_0\vert _{\partial B_1}\in H^{3/2}(\partial B_1)$. The point of our uniqueness result is that no boundary regularity assumption is needed. Second, we prove the exponential convergence of the harmonic map heat flow, assuming that energy is small at all times.


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Additional Information

Lu Wang
Affiliation: Department of Mathematics, Massachusetts Institute of Technology, 77 Massachusetts Avenue, Cambridge, Massachusetts 02139
Address at time of publication: Department of Mathematics, Johns Hopkins University, 3400 N. Charles Street, Baltimore, Maryland 21218
Email: luwang@math.mit.edu, lwang@math.jhu.edu

DOI: http://dx.doi.org/10.1090/S0002-9947-2012-05473-0
PII: S 0002-9947(2012)05473-0
Received by editor(s): July 22, 2010
Received by editor(s) in revised form: September 11, 2010
Published electronically: May 29, 2012
Article copyright: © Copyright 2012 American Mathematical Society
The copyright for this article reverts to public domain 28 years after publication.