论文标题
最小$ W^{
Minimal $W^{s,\frac{n}{s}}$-harmonic maps in homotopy classes
论文作者
论文摘要
令$σ$ a封闭$ n $ dimensional歧管，$ \ mathcal {n} \ subset \ mathbb {r}^m $是一个封闭的歧管，而$ u \ in w^{s，\ frac ns}（σ，=，\ mathcal {n}）$ for $ s in（0,0,1）$。我们通过证明$π_n（\ Mathcal {n}）= \ {0 \ {0 \} $，扩展了麻袋和Uhlenbeck的巨大作品，则存在一个最小化$ W^{s，\ frac ns} $ - harmonic Map harmonic Map同型对$ u $。如果$π_n（\ Mathcal {n}）\ neq \ {0 \} $，那么我们证明存在$ w^{s，\ frac {n} {n} {s}}} $ - harmonic map trof $ \ \ \ \ \ \ \ \ \ m mathbb {s}^n $ to $ \ n $ $π_{n}（\ Mathcal {n}）$。 由于在分数框架中未知几种技术，尤其是pohozaev型论点（尤其是当$ \ frac {n} {n} {s} {s} \ neq 2 $时，一个人无法通过扩展方法争论），我们会开发出重要的新工具，这些新工具自身很有趣：例如，对于点数而不是重新启动的能力和平衡的能量，并且可以进行平衡的能量估算。此外，我们证明了最小化$ w^{s，\ frac {n} {s}} $ - 映射到歧管中的规律性理论。
Let $Σ$ a closed $n$-dimensional manifold, $\mathcal{N} \subset \mathbb{R}^M$ be a closed manifold, and $u \in W^{s,\frac ns}(Σ,\mathcal{N})$ for $s\in(0,1)$. We extend the monumental work of Sacks and Uhlenbeck by proving that if $π_n(\mathcal{N})=\{0\}$ then there exists a minimizing $W^{s,\frac ns}$-harmonic map homotopic to $u$. If $π_n(\mathcal{N})\neq \{0\}$, then we prove that there exists a $W^{s,\frac{n}{s}}$-harmonic map from $\mathbb{S}^n$ to $\mathcal{N}$ in a generating set of $π_{n}(\mathcal{N})$. Since several techniques, especially Pohozaev-type arguments, are unknown in the fractional framework (in particular when $\frac{n}{s} \neq 2$ one cannot argue via an extension method), we develop crucial new tools that are interesting on their own: such as a removability result for point-singularities and a balanced energy estimate for non-scaling invariant energies. Moreover, we prove the regularity theory for minimizing $W^{s,\frac{n}{s}}$-maps into manifolds.
