Characterizing Two-Dimensional Maps Whose Jacobians Have Constant Eigenvalues

Abstract

AbstractRecent papers have shown that C1 maps 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 formfor some constants a, b, c, d, e, f , α, β and a C1 function ϕ in one variable. If, in addition, the function ϕ is not affine, thenThis paper shows how these theorems cannot be extended by constructing a real-analytic map whose Jacobian eigenvalues are ±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.