球面调和映射的Lavrentiev空隙现象在稠密的零次边界数据集上成立
The Lavrentiev gap phenomenon for harmonic maps into spheres holds on a dense set of zero degree boundary data
摘要：
本文证明了对每个正整数$N$，光滑零次映射$ψ\colon\mathbb{S}^2\to \mathbb{S}^2$的集合具有以下三个性质：（1）在单位球的边界上存在唯一的极小调和映射$u\colon \mathbb{B}^3\to \mathbb{S}^2$，它与$ψ$一致;（2）这个映射$u$在$\mathbb{B}^3$中至少有$N$奇点;（3）Lavrentiev空隙现象对$ψ$成立，即，$ψ$的所有光滑扩张$w\colon \mathbb{B}^3\to\mathbb{S}^2$的Dirichlet能量$E(w)$的下确界严格大于（不规则）极小元$u$的Dirichlet能量$\int_{\mathbb{B}^3} |\nabla u|^2$，在赋予$W^{1,p}$ -拓扑的所有光滑零次映射$ϕ\colon \mathbb{S}^2\to\mathbb{S}^2$的集合中是稠密的，其中$1\le p < 2$ .这个结果是尖锐的：它在所有光滑边界数据集上的$W^{1,2}$拓扑中失败。
英文摘要：
We prove that for each positive integer $N$ the set of smooth, zero degree maps $ψ\colon\mathbb{S}^2\to \mathbb{S}^2$ which have the following three properties: (1) there is a unique minimizing harmonic map $u\colon \mathbb{B}^3\to \mathbb{S}^2$ which agrees with $ψ$ on the boundary of the unit ball; (2) this map $u$ has at least $N$ singular points in $\mathbb{B}^3$; (3) the Lavrentiev gap phenomenon holds for $ψ$, i.e., the infimum of the Dirichlet energies $E(w)$ of all smooth extensions $w\colon \mathbb{B}^3\to\mathbb{S}^2$ of $ψ$ is strictly larger than the Dirichlet energy $\int_{\mathbb{B}^3} |\nabla u|^2$ of the (irregular) minimizer $u$, is dense in the set of all smooth zero degree maps $ϕ\colon \mathbb{S}^2\to\mathbb{S}^2$ endowed with the $W^{1,p}$-topology, where $1\le p < 2$. This result is sharp: it fails in the $W^{1,2}$ topology on the set of all smooth boundary data.
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。
