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# Characterizing Two-Dimensional Maps Whose Jacobians Have Constant Eigenvalues

Published:2003-09-01
Printed: Sep 2003
• Marc Chamberland
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## Abstract

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 $$\alpha\beta (d-a) + b\alpha^2 - c\beta^2 = 0.$$ 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--Ampère equation
 MSC Classifications: 26B10 - Implicit function theorems, Jacobians, transformations with several variables 14R15 - Jacobian problem [See also 13F20] 35L70 - Nonlinear second-order hyperbolic equations