论文标题
Sobolev映射的痕量理论成分
Trace theory for Sobolev mappings into a manifold
论文作者
论文摘要
我们回顾了有关空间痕迹$ w^{1，p}（\ Mathbb {b}^{M-1} \ times（0,1），\ Mathcal {n}）$ sobolev mappings toble compact compact cormold $ \ m m mathcal {n} $的目前状态。特别是，我们表现出对扩展的新分析障碍，当$ p <m $是一个整数时，并且同型组$π_p（\ mathcal {n}）$是无琐的。从积极的一面来看，当基本组$π_1（\ MATHCAL {n}）$是有限的，$π_2（\ Mathcal {n}）\ simeq \ dotsb \ dotsb \simeqπ_{ \ {0 \} $。我们提出与扩展问题有关的几个开放问题。
We review the current state of the art concerning the characterization of traces of the spaces $W^{1, p} (\mathbb{B}^{m-1}\times (0,1), \mathcal{N})$ of Sobolev mappings with values into a compact manifold $\mathcal{N}$. In particular, we exhibit a new analytical obstruction to the extension, which occurs when $p < m$ is an integer and the homotopy group $π_p (\mathcal{N})$ is non trivial. On the positive side, we prove the surjectivity of the trace operator when the fundamental group $π_1 (\mathcal{N})$ is finite and $π_2 (\mathcal{N}) \simeq \dotsb \simeq π_{\lfloor p - 1 \rfloor} (\mathcal{N}) \simeq \{0\}$. We present several open problems connected to the extension problem.
