Problem of Functional Extension and Space-Like Surfaces in Minkowski Space

Abstract

Let $\Xi (x)$ be the distribution of convex sets over a domain $D\subset {\mathbb{R}}^n$ and let $\varphi :\partial D\to \mathbb{R}$ be a function. We consider the existence problem of locally Lipschitz functions $f$ defined in the domain $D$ so that $f{|}_{\partial D}=\varphi$ and $▽f(x)\in \Xi (x)$ almost everywhere in $D$. These questions are related to the existence problem for space-like surfaces of arbitrary codimension with prescribed boundary in Minkowski space.