有限Hessian能量映射的序列弱逼近
Sequential Weak Approximation for Maps of Finite Hessian Energy
摘要：
考虑从光滑域$Ω\subset\R^m$到Hessian能量$\int_Ω|\nabla^2 v|^2\, dx$有限的紧致黎曼流形$N$的二阶Sobolev映射$\ v\ $的空间$W^{2,2}(Ω;N)$.本文研究了$N$的拓扑与一个$W^{2,2}$映射的$W^{2,2}$强或弱可逼近性之间的关系。我们详细处理$W^{2,2}(\B^5,S^3)$，其中我们建立了$W^{2,2}(\B^5,S^3)\cap{\mathcal C}^\infty$的\underline{sequential weak} $W^{2,2}$密度。高阶Sobolev映射的强\cite{BPV}逼近性在P.Bousquet，A. Ponce，and J. Van Schaftigen.对于一个单独的地图$v\in W^{2,2}(\B^5,S^3)$，我们定义了一个数$L(v)$，它近似是连接$v$的强逼近$u$的孤立奇点彼此或$\p\B^5$所需的总长度。则$L(v)=0$当且仅当$v$容许$W^{2,2}$被光滑映射强逼近.我们的关键结果，通过构建特定的曲线连接的奇异性$u$，是绑定$\ L(u)\leq c\int_{\B^5}|\nabla^2 u|^2\, dx\ $。这使我们能够构造，对于给定的Sobolev映射$v\in W^{2,2}(\B^5,S^3)$，所需的$W^{2,2}$弱逼近序列的光滑映射。为了找到$u$的合适连接曲线，使用$u$的合适水平面的$u$拉回法向框架的扭曲
英文摘要：
Consider the space $W^{2,2}(Ω;N)$ of second order Sobolev mappings $\ v\ $ from a smooth domain $Ω\subset\R^m$ to a compact Riemannian manifold $N$ whose Hessian energy $\int_Ω|\nabla^2 v|^2\, dx$ 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 $Ω$ 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 $u$
机构、总结、中文摘要均由AI分析而来，可能存在错误，请以论文正文为准。
