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 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.