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 $\phi: \partial D \rightarrow \mathbb R$ be a function. We consider the existence problem of locally Lipschitz functions $f$ defined in the domain $D$ so that $f|_{\partial D} = \phi$ and $\bigtriangledown f (x) \in \Xi (x)$ 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.