A non-smooth map with smooth Jacobian?

Question

I am looking for an example of a non-smooth Sobolev map $f \in W_{\text{loc}}^{1,d}(\Omega,\mathbb{R}^d)$, where $\Omega$ is an open subset of $\mathbb{R}^d$, such that $\det df \in C^{\infty}$ is everywhere non-zero.

In other words, I am looking for a map which is less regular that its Jacobian.

Answer 1

For example, a shear deformation $f:\mathbb{R}^2\to\mathbb{R}^2$ $$ f(x, y) = (x, y + h(x)) $$ where $h$ is a Sobolev function. The Jacobian determinant is identically $1$.