On the Reconstruction Problem for Nondifferentiable Functions

Abstract

In this paper we consider the following reconstruction problem: Let $U \subset {\mathbb{R}^n}$ be an open set and let $f:U \to P\left( {{{\mathbb{R}}^{n}}} \right)$ be a function with values in $P\left( {{\mathbb{R}^n}} \right)$, the space of positively homogeneous functions on ${\mathbb{R}^n}$. Does there exist a directionally-differentiable locally Lipschitz function $F:U \to \mathbb{R}$, such that the directional derivative dF|_x coincides with f (x) for all x ∈ U ? How works a construction of F ?

To investigate these questions we will essentially make use of a special representation of f.

Based on this representation generalizations of classical means like tangential vector, tangential space and differential forms will be introduced. They turn out to be helpful tools to give answers to the reconstruction problem for special cases by providing a nonsmooth variant of the classical Lemma of Poincaré.