Partial regularity of a minimizer of the relaxed energy for biharmonic maps

In this paper, we study the relaxed energy for biharmonic maps from an m-dimensional domain into spheres for an integer m ⩾ 5 . By an approximation method, we prove the existence of a minimizer of the relaxed energy of the Hessian energy, and that the minimizer is biharmonic and smooth outside a sin...

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Bibliographic Details
Published in:Journal of functional analysis 2012-01, Vol.262 (2), p.682-718
Main Authors: Hong, Min-Chun, Yin, Hao
Format: Article
Language:English
Subjects:
Biharmonic maps
Partial regularity
Relaxed energy
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Summary:In this paper, we study the relaxed energy for biharmonic maps from an m-dimensional domain into spheres for an integer m ⩾ 5 . By an approximation method, we prove the existence of a minimizer of the relaxed energy of the Hessian energy, and that the minimizer is biharmonic and smooth outside a singular set Σ of finite ( m − 4 ) -dimensional Hausdorff measure. When m = 5 , we prove that the singular set Σ is 1-rectifiable. Moreover, we also prove a rectifiability result for the concentration set of a sequence of stationary harmonic maps into manifolds.
ISSN:0022-1236
1096-0783
DOI:10.1016/j.jfa.2011.10.003