Characterizing Two-Dimensional Maps Whose Jacobians Have Constant Eigenvalues

[摘要] Recent papers have shown that $C^1$ maps $Fcolon mathbb{R}^2ightarrow mathbb{R}^2$whose Jacobians have constant eigenvalues can be completelycharacterized if either the eigenvalues are equal or $F$ is apolynomial. Specifically, $F=(u,v)$ must take the formegin{gather*}u = ax + by + eta phi(alpha x + eta y) + e \v = cx + dy - alpha phi(alpha x + eta y) + fend{gather*}for some constants $a$, $b$, $c$, $d$, $e$, $f$, $alpha$, $eta$ anda $C^1$ function $phi$ in one variable. If, in addition, the function$phi$ is not affine, thenegin{equation}alphaeta (d-a) + balpha^2 - ceta^2 = 0.end{equation}This paper shows how these theorems cannot be extended by constructinga real-analytic map whose Jacobian eigenvalues are $pm 1/2$ and doesnot fit the previous form. This example is also used to constructnon-obvious solutions to nonlinear PDEs, including the Monge--Amp`ereequation.

[学科分类] 数学（综合）

[关键词] Jacobian Conjecture;injectivity;Monge--Ampère equation