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 $W^{2,2}(\mathbb {B}^5,\mathbb {S}^3)$ where we establish the sequential weak $W^{2,2}$ density of $W^{2,2}(\mathbb {B}^5,\mathbb {S}^3)\,\cap \,{\mathcal C}^\infty$. The strong $W^{2,2}$ approximability of higher order Sobolev maps has been studied in the recent paper of Bousquet et al. (J. Eur. Math. Soc. (JEMS) 17(4), 763–817, 2015). For an individual map $v\in W^{2,2}(\mathbb {B}^5,\mathbb {S}^3)$, 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 $\partial Bx^5$. 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)\le c+c\int _{\mathbb {B}^5}|\nabla ^2 u|^2\, dx\$. This allows us to construct, for the given Sobolev map $v\in W^{2,2}(\mathbb {B}^5,\mathbb {S}^3)$, 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 on a suitable level surface of u.