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 $\mathrm{\Omega }\subset R^n$ 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 $\lambda\in [0,1]$weintroducea$\lambda$-relaxedenergy$\H_\lambda$ for the Hessian energy for maps in $W^{2,2}(\mathrm{\Omega },S^4)$ 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 $\lambda \in [0,1)$.