Expand all Collapse all | Results 1 - 2 of 2 |
1. CMB 2003 (vol 46 pp. 323)
Characterizing Two-Dimensional Maps Whose Jacobians Have Constant Eigenvalues Recent papers have shown that $C^1$ maps $F\colon \mathbb{R}^2
\rightarrow \mathbb{R}^2$
whose Jacobians have constant eigenvalues can be completely
characterized if either the eigenvalues are equal or $F$ is a
polynomial. Specifically, $F=(u,v)$ must take the form
\begin{gather*}
u = ax + by + \beta \phi(\alpha x + \beta y) + e \\
v = cx + dy - \alpha \phi(\alpha x + \beta y) + f
\end{gather*}
for some constants $a$, $b$, $c$, $d$, $e$, $f$, $\alpha$, $\beta$ and
a $C^1$ function $\phi$ in one variable. If, in addition, the function
$\phi$ is not affine, then
\begin{equation}
\alpha\beta (d-a) + b\alpha^2 - c\beta^2 = 0.
\end{equation}
This paper shows how these theorems cannot be extended by constructing
a real-analytic map whose Jacobian eigenvalues are $\pm 1/2$ and does
not fit the previous form. This example is also used to construct
non-obvious solutions to nonlinear PDEs, including the Monge--Amp\`ere
equation.
Keywords:Jacobian Conjecture, injectivity, Monge--AmpÃ¨re equation Categories:26B10, 14R15, 35L70 |
2. CMB 1998 (vol 41 pp. 442)
A Mountain Pass to the Jacobian Conjecture. This paper presents an approach to injectivity theorems via the
Mountain Pass Lemma and raises an open question. The main result
of this paper (Theorem~1.1) is proved by means of the Mountain Pass
Lemma and states that if the eigenvalues of $F' (\x)F' (\x)^{T}$
are uniformly bounded away from zero for $\x \in \hbox{\Bbbvii
R}^{n}$, where $F \colon \hbox{\Bbbvii R}^n \rightarrow
\hbox{\Bbbvii R}^n$ is a class $\cC^{1}$ map, then $F$ is
injective. This was discovered in a joint attempt by the authors
to prove a stronger result conjectured by the first author: Namely,
that a sufficient condition for injectivity of class $\cC^{1}$ maps
$F$ of $\hbox{\Bbbvii R}^n$ into itself is that all the eigenvalues
of $F'(\x)$ are bounded away from zero on $\hbox{\Bbbvii
R}^n$. This is stated as Conjecture~2.1. If true, it would imply
(via {\it Reduction-of-Degree}) {\it injectivity of polynomial
maps} $F \colon \hbox{\Bbbvii R}^n \rightarrow \hbox{\Bbbvii R}^n$
{\it satisfying the hypothesis}, $\det F'(\x) \equiv 1$, of the
celebrated Jacobian Conjecture (JC) of Ott-Heinrich Keller. The
paper ends with several examples to illustrate a variety of cases
and known counterexamples to some natural questions.
Keywords:Injectivity, ${\cal C}^1$-maps, polynomial maps, Jacobian Conjecture, Mountain Pass Categories:14A25, 14E09 |