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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 |