Sequential Weak Approximation for Maps of Finite Hessian Energy

Consider the space $W^{2,2}(\mathrm{\Omega };N)$ of second order Sobolev mappings \ v\ from a smooth domain $\mathrm{\Omega }\subset {\mathbb{R}}^m$ to a compact Riemannian manifold $N$ whose Hessian energy ${\int }_{\mathrm{\Omega }}\mathrm{\mid }{\mathrm{\nabla }}^{2}v{\mathrm{\mid }}^{2}dx$ is finite. Here we are interested in relations between the topology of $N$ and the $W^{2,2}$ strong or weak approximability of a $W^{2,2}$ map by a sequence of smooth maps from $\mathrm{\Omega }$ to $N$. We treat in detail $W^{2,2}({\text{B}}^5,S^3)$ where we establish the \underline{sequential weak} $W^{2,2}$ density of $W^{2,2}({\text{B}}^5,S^3)\cap {\mathcal{C}}^{\mathrm{\infty }}$. The strong $W^{2,2}$ approximability of higher order Sobolev maps has been studied in the recent preprint \cite{BPV} of P. Bousquet, A. Ponce, and J. Van Schaftigen. For an individual map $v\in W^{2,2}({\text{B}}^5,S^3)$, we define a number $L(v)$ which is approximately the total length required to connect the isolated singularities of a strong approximation $u$ of $v$ either to each other or to $\text{p}{\text{B}}^5$. Then $L(v)=0$ if and only if $v$ admits $W^{2,2}$ strongly approximable by smooth maps. Our critical result, obtained by constructing specific curves connecting the singularities of $u$, is the bound $\ L(u)\leq c\int_{\B^5}|\nabla^2 u|^2, dx\ $\mathrm{.}Thisallowsustoconstruct,forthegivenSobolevmap$v\in W^{2,2}(\B^5,S^3)$,thedesired$W^{2,2}$ weakly approximating sequence of smooth maps. To find suitable connecting curves for $u$, one uses the twisting of a $u$ pull-back normal framing of a suitable level surface of $u$