论文标题
单点Reshetnyak的定理
A single-point Reshetnyak's theorem
论文作者
论文摘要
我们证明了Reshetnyak定理的单值版本。也就是说，如果从w^{1，n} _ {\ text {loc}}（ω，\ mathbb {r}^n）$ in W^{1，n} _ {1，n} _ {1，n} _ {1，n} _ {r}^n）$中σ（x）\ lvert f（x）-y_0 \ rvert^n $ for Some $ k \ geq 1 $，$ y_0 \ in \ Mathbb {r}^n $和$σ\ in l^{1+ \ varepsilon} _ {1+\ varepsilon} _ { $ f^{ - 1} \ {y_0 \} $是离散的，本地索引$ i（x，f）$在$ f^{ - 1} \ {y_0 \} $中是阳性的，$ f^{ - 1} \ {y_0 \} $的每个社区均为$ y__的$ y_0 $。假设在\ mathbb {r}^n $中的每个$ y_0 \ in y Mathbb {r}^n $中的固定$ k $的估计值相当于假设地图$ f $是$ k $ -quasiregular，即使每个$ y__0 $的选择都是不同的。由于该估计值还产生单值liouville定理，因此它似乎是$ k $ quasiregularity的一个很好的定义。作为我们的单值Reshetnyak定理的推论，我们获得了参数原理的较高维度版本，该版本在解决Calderón问题的解决方案中起着关键作用。
We prove a single-value version of Reshetnyak's theorem. Namely, if a non-constant map $f \in W^{1,n}_{\text{loc}}(Ω, \mathbb{R}^n)$ from a domain $Ω\subset \mathbb{R}^n$ satisfies the estimate $\lvert Df(x) \rvert^n \leq K J_f(x) + Σ(x) \lvert f(x) - y_0 \rvert^n $ for some $K \geq 1$, $y_0\in \mathbb{R}^n$ and $Σ\in L^{1+\varepsilon}_{\text{loc}}(Ω)$, then $f^{-1}\{y_0\}$ is discrete, the local index $i(x, f)$ is positive in $f^{-1}\{y_0\}$, and every neighborhood of a point of $f^{-1}\{y_0\}$ is mapped to a neighborhood of $y_0$. Assuming this estimate for a fixed $K$ at every $y_0 \in \mathbb{R}^n$ is equivalent to assuming that the map $f$ is $K$-quasiregular, even if the choice of $Σ$ is different for each $y_0$. Since the estimate also yields a single-value Liouville theorem, it hence appears to be a good pointwise definition of $K$-quasiregularity. As a corollary of our single-value Reshetnyak's theorem, we obtain a higher-dimensional version of the argument principle that played a key part in the solution to the Calderón problem.
