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 singular set Sigma of finite (m - 4)-dimensional Hausdorff measure. When m = 5, we prove that the singular set Sigma is I-rectifiable. Moreover, we also prove a rectifiability result for the concentration set of a sequence of stationary harmonic maps into manifolds. Crown Copyright (C) 2011 Published by Elsevier Inc. All rights reserved.