论文标题
有限失真的单调图的纤维
Fibers of monotone maps of finite distortion
论文作者
论文摘要
我们研究了拓扑单调的汇总$ w^{1，n} $ - 有限失真$ f \ colonω\ toω'$，其中$ω，ω'$是$ \ mathbb {r}^n $，$ n \ egeq 2 $的域中的域。如果外部失真函数$ k_f \ in l _ {\ mathrm {loc}}}^{p}（ω）$带有$ p \ geq n-1 $，则已知任何此类地图$ f $都是同源的，因此fibers $ f^{ - 1}}} \ {y \ \ {y \ \ {y singletons singletons singletons。我们表明，作为失真函数的集成性$ p $的指数，$ k_f $在$ 1/（n-1）\ leq p <n-1 $中增加，然后纤维$ f^{ - 1} \ {y \} $ of $ f $开始满足越来越强大的同源限制。我们还通过单调$ f \ colon \ mathbb {r}^3 \ to \ mathbb {r}^3 $带有同源性非平凡的纤维的单调$ f \ colon \ mathbb {r}^3 \ to sobolev实现了拓扑示例，并证明此示例具有$ k_f \ in l^{1/2----------------------------- \ varepsilon} _ {\ mathrm {loc}}（\ mathbb {r}^3）$ for ALL $ \ VAREPSILON> 0 $。
We study topologically monotone surjective $W^{1,n}$-maps of finite distortion $f \colon Ω\to Ω'$, where $Ω, Ω' $ are domains in $\mathbb{R}^n$, $n \geq 2$. If the outer distortion function $K_f \in L_{\mathrm{loc}}^{p}(Ω)$ with $p \geq n-1$, then any such map $f$ is known to be homeomorphic, and hence the fibers $f^{-1}\{y\}$ are singletons. We show that as the exponent of integrability $p$ of the distortion function $K_f$ increases in the range $1/(n-1) \leq p < n-1$, then the fibers $f^{-1}\{y\}$ of $f$ start satisfying increasingly strong homological limitations. We also give a Sobolev realization of a topological example by Bing of a monotone $f \colon \mathbb{R}^3 \to \mathbb{R}^3$ with homologically nontrivial fibers, and show that this example has $K_f \in L^{1/2 - \varepsilon}_{\mathrm{loc}}(\mathbb{R}^3)$ for all $\varepsilon > 0$.
