On some nonlinear expressions of the Jacobian

Abstract

We define the expressions $J\log^{1+\alpha}(e+|D f|)$ and $J\log^{1+\alpha}(e+|J|)$ as Schwartz distributions, for $f\colon\Omega\subset{{\Bbb R}}^n\to {{\Bbb R}}^n$ a Sobolev mapping such that $|D f|^n\log^\alpha(e+|D f|)$ is locally integrable, $-1 <\alpha <0$, and $J$ the Jacobian determinant.