Sequential Weak Approximation for Maps of Finite Hessian Energy

Abstract

Consider the space $W^{2,2}(\Omega ;N)$ of second order Sobolev mappings $v$ from a smooth domain $\Omega \subset {\mathbb{R}}^m$ to a compact Riemannian manifold $N$ whose Hessian energy ${\int }_{\Omega }|{\nabla }^{2}v{|}^{2}dx$ 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 $\Omega$ to $N$. We treat in detail $W^{2,2}(\B^5,S^3)$ where we establish the \underline{sequential weak} $W^{2,2}$ density of $W^{2,2}(\B^5,S^3)\cap{\mathcal C}^\infty$. 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 $v\in W^{2,2}(\B^5,S^3)$, 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 $\p\B^5$. 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\ $. This allows us to construct, for the given Sobolev map $v\in W^{2,2}(\B^5,S^3)$, the desired $W^{2,2}$ 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 $u$