临界情形下有限畸变映射的离散性和开性$p=n-1$
Discreteness and openness for mappings of finite distortion in the critical case $p=n-1$
摘要：
设$F\in W_{loc}^{1,n}(Ω;\Bbb R^n)$是具有非负雅可比矩阵$J_F(x)=\text{det} DF(x)\ge 0$ a.e.在域$Ω\in \Bbb R^n$中。映射$F$的扩张几乎在$Ω$中的任何地方都由公式$$K(x)={{|DF(x)|^n}\over {J_F(x)}}.$$定义。如果$K(x)$有界a.e.，该映射称为拟正则映射。拟正则映射是高维全纯映射的推广。高维拟正则映射的理论始于Rešhetnyak定理，该定理指出非常数拟正则映射是连续的、离散的和开的。在非线性弹性理论的某些问题中，$K(x)$的有界性条件限制太大。典型地，我们只知道$F$具有有限扩张，即$K(x)$是有限a.e.的。并且对于某个值$p$，$K(x)^p$是可积的。在二维空间中，Iwaniec和Šverak [IS]证明了$K(x)\in L^1_{loc}$足以保证Rešhetnyak定理的结论。对于$n\ge 3$，Heinonen和Kotina [HK]证明了如果映射是拟轻的，而$K(x)\in L^p_{loc}$对于$p>n-1$，则映射$F(x)$是连续的、离散的和开的. Manfredi和Villamor [MV]证明了一个类似的结果，而没有假设映射$f(x)$是准轻的。已知结果为假，参见[球]，当$p<n-1$。在本文中，我们试图改善这些结果。特别地，我们将处理对于$n\ge 3$的$p=n-1$的情况，并且将假设我们的映射$F(x)$是准轻的，即在$Ω$中任何点的逆像是紧的。我们的方法将不同于[MV]和[HK]中使用的方法。它在本质上更具有几何性，并使用极值长度法。
英文摘要：
Let $F\in W_{loc}^{1,n}(Ω;\Bbb R^n)$ be a mapping with non-negative Jacobian $J_F(x)=\text{det} DF(x)\ge 0$ a.e. in a domain $Ω\in \Bbb R^n$. The dilatation of the mapping $F$ is defined, almost everywhere in $Ω$, by the formula $$K(x)={{|DF(x)|^n}\over {J_F(x)}}.$$ If $K(x)$ is bounded a.e., the mapping is said to be quasiregular. Quasiregular mappings are a generalization to higher dimensions of holomorphic mappings. The theory of higher dimensional quasiregular mappings began with Rešhetnyak's theorem, stating that non constant quasiregular mappings are continuous, discrete and open. In some problems appearing in the theory of non-linear elasticity, the boundedness condition on $K(x)$ is too restrictive. Tipically we only know that $F$ has finite dilatation, that is, $K(x)$ is finite a.e. and $K(x)^p$ is integrable for some value $p$. In two dimensions, Iwaniec and Šverak [IS] have shown that $K(x)\in L^1_{loc}$ is sufficient to guarantee the conclusion of Rešhetnyak's theorem. For $n\ge 3$, Heinonen and Koskela [HK], showed that if the mapping is quasi-light and $K(x)\in L^p_{loc}$ for $p>n-1$, then the mapping $F(x)$ is continuous, discrete and open. Manfredi and Villamor [MV] proved a similar result without assuming that the mapping $f(x)$ was quasi-light. The result is known to be false, see [Ball], when $p<n-1$. In this paper we attempt to improve in those results. In particular, we will deal with the case $p=n-1$ for $n\ge 3$, and will assume that our mapping $F(x)$ is quasi-light, that is, the inverse image of any point is compact in $Ω$. Our approach will be different from the ones used in [MV] and [HK]. It is more geometrical in nature and uses the method of extremal length.
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