Regularity and relaxed problems of minimizing biharmonic maps into spheres

For $n\ge 5$ and $k\ge 4$, we show that any minimizing biharmonic map from $\Omega\subset R^n$ to $S^k$ is smooth off a closed set whose Hausdorff dimension is at most $n-5$. When $n=5$ and $k=4$, for a parameter $\lambda\in [0,1]$ we introduce a $\lambda$-relaxed energy $\H_\lambda$ for the Hessian energy for maps in $W^{2,2}(\Omega,S^4)$ so that each minimizer $u_\lambda$ of $\H_\lambda$ is also a biharmonic map. We also estabilish the existence and partial regularity of a minimizer of $\H_\lambda$ for $\lambda\in [0,1)$.