论文标题
平滑映射的映射，最多$ 1 $的衍生物等级
Smooth approximation of mappings with rank of the derivative at most $1$
论文作者
论文摘要
有人指出，如果$ f \在c^1（\ mathbb {r}^n，\ mathbb {r}^n）$满足$ \ perperatorName {rank} df \ leq m <n $ wherewhere wherewhere $ \ \ \ \ m m mathbb {r}^n $，然后由$ f $ app $ c $ c $ c. $ \ operatatorName {rank} dg \ leq m $无处不在。总的来说，虽然对此猜想有反例，但我们证明当$ m = 1 $时，答案为正。更确切地说，如果$ m = 1 $，我们的结果会产生局部lipschitz映射$ f：ω\ to \ mathbb {r}^n $的几乎均匀的近似1 $，提供的$ω\ subset \ mathbb {r}^n $仅连接。近似值的构建采用了公制空间的分析技术，包括公制树理论（$ \ mathbb {r} $ - 树）。
It was conjectured that if $f\in C^1(\mathbb{R}^n,\mathbb{R}^n)$ satisfies $\operatorname{rank} Df\leq m<n$ everywhere in $\mathbb{R}^n$, then $f$ can be uniformly approximated by $C^\infty$-mappings $g$ satisfying $\operatorname{rank} Dg\leq m$ everywhere. While in general, there are counterexamples to this conjecture, we prove that the answer is in the positive when $m=1$. More precisely, if $m=1$, our result yields an almost-uniform approximation of locally Lipschitz mappings $f:Ω\to\mathbb{R}^n$, satisfying $\operatorname{rank} Df\leq 1$ a.e., by $C^\infty$-mappings $g$ with $\operatorname{rank} Dg\leq 1$, provided $Ω\subset\mathbb{R}^n$ is simply connected. The construction of the approximation employs techniques of analysis on metric spaces, including the theory of metric trees ($\mathbb{R}$-trees).
