具有指定特征向量的雅可比矩阵
Jacobians with prescribed eigenvectors
摘要：
令$Ω\subset \mathbb{R}^n$是开的，并且令$\mathcal{R}$是$Ω$上的部分帧，即在$Ω$上规定的一组$m$线性无关向量场（$m\leq n$）。我们考虑的问题描述的所有地图$F:Ω\to\mathbb{R}^n$的属性，每个给定的向量场是一个特征向量的雅可比矩阵的$F$的集合。通过引入雅可比矩阵的坐标独立定义，我们得到了问题的内在形式，从而得到了一个超定偏微分方程系统，其相容性条件可以用内在的、坐标独立的方式表示.为了分析这个系统，我们制定并证明了一个推广的经典Frobenius可积定理。该系统解集的大小和结构取决于部分标架的性质，特别是它是否处于对合中。对合部分框架的一个特别好的子类，称为rich，可以完全分析。对合但非富的情况有点难以处理。我们提供了一个完整的答案的情况下，$m=3$和任意$n$，以及一些一般性的结果，任意$m$。非对合的情况更具挑战性，我们只在$n=3$，$m=2$的情况下获得全面的分析。最后，我们提供了明确的例子说明各种可能性。我们考虑这个问题的最初动机来自于一维双曲保守系统的几何研究。
英文摘要：
Let $Ω\subset \mathbb{R}^n$ be open and let $\mathcal{R}$ be a partial frame on $Ω$, that is a set of $m$ linearly independent vector fields prescribed on $Ω$ ($m\leq n$). We consider the issue of describing the set of all maps $F:Ω\to\mathbb{R}^n$ with the property that each of the given vector fields is an eigenvector of the Jacobian matrix of $F$. By introducing a coordinate independent definition of the Jacobian, we obtain an intrinsic formulation of the problem, which leads to an overdetermined PDE system, whose compatibility conditions can be expressed in an intrinsic, coordinate independent manner. To analyze this system we formulate and prove a generalization of the classical Frobenius integrability theorems. The size and structure of the solution set of this system depends on the properties of the partial frame, in particular, whether or not it is in involution. A particularly nice subclass of involutive partial frames, called rich, can be completely analyzed. Involutive, but non-rich case is somewhat harder to handle. We provide a complete answer in the case of $m=3$ and arbitrary $n$, as well as some general results for arbitrary $m$. The non-involutive case is far more challenging, and we only obtain a comprehensive analysis in the case $n=3$, $m=2$. Finally, we provide explicit examples illustrating the various possibilities. Our initial motivation for considering this problem comes from the geometric study of hyperbolic conservative systems in one spatial dimension.
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