有限畸变单调映射的纤维
Fibers of monotone maps of finite distortion
摘要：
研究了有限畸变$f \colon Ω\to Ω'$的拓扑单调满射$W^{1,n}$ -映射，其中$Ω, Ω' $是$\mathbb{R}^n$，$n \geq 2$中的整环.如果外畸变函数$K_f \in L_{\mathrm{loc}}^{p}(Ω)$具有$p \geq n-1$，则已知任何这样的映射$f$是同胚的，因此纤维$f^{-1}\{y\}$是单例的。我们表明，作为可积性$p$的失真函数$K_f$的指数增加的范围$1/(n-1) \leq p < n-1$，然后纤维$f^{-1}\{y\}$的$f$开始满足越来越强的同调限制。我们还给出了一个Sobolev实现的一个拓扑的例子，由冰的单调$f \colon \mathbb{R}^3 \to \mathbb{R}^3$同调非平凡纤维，并表明，这个例子有$K_f \in L^{1/2 - \varepsilon}_{\mathrm{loc}}(\mathbb{R}^3)$的所有$\varepsilon > 0$。
英文摘要：
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 $\varepsilon > 0$.
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