Characterizing Two-Dimensional Maps Whose Jacobians Have Constant Eigenvalues

Chamberland, Marc

Canadian mathematical bulletin, Tome 46 (2003) no. 3, pp. 323-331

Résumé

Recent papers have shown that ${{C}^{1}}$ maps $F:\,{{\mathbb{R}}^{2}}\,\to {{\mathbb{R}}^{2}}$ 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 $$u\,=\,ax\,+\,by\,+\,\beta \phi (\alpha x\,+\,\beta y)\,+\,e$$ $$v\,=\,cx\,+\,dy\,-\,\alpha \phi \,(\alpha x\,+\,\beta y)\,+\,f$$ 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 1 $$\alpha \beta (d\,-\,a)\,+\,b{{\alpha }^{2}}\,-\,c{{\beta }^{2}}\,=\,0.$$ 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—Ampère equation.

Détail

DOI : 10.4153/CMB-2003-034-4

Mots-clés : 26B10, 14R15, 35L70, Jacobian Conjecture, injectivity, Monge—Ampère equation