A geometrical approach to monotone functions in $\mathbb R^n$

Abstract.

The paper is concerned with the fine properties of monotone functions on $\mathbb{R}^n$. We study the continuity and differentiability properties of these functions, the approximability properties, the structure of the distributional derivatives and of the weak Jacobians. Moreover, we exhibit an example of a monotone function u which is the gradient of a $C^{1,\alpha}$ convex function and whose weak Jacobian Ju is supported on a purely unrectifiable set.