Problem of functional extension and space-like surfaces in Minkowski space
Let \(\Theta\) be a distribution of convex sets over a domain \(D \subset \mathbb{R}^n\) and \(\Phi : \partial D \rightarrow \mathbb{R}\) be a function. The authors consider the problem of the existence of locally Lipschitz functions \(f : D \rightarrow \mathbb{R}\) such that \(f|_{\partial D}= \Phi\) and \(\nabla f(x) \in \Theta (x)\) almost everywhere in \(D\). A criterion for the solvability of this problem is given, in the case of locally bounded convex sets in \(\Theta\). The results are applied to the existence problem of \(k\)-dimensional spacelike surfaces with prescribed boundary in the Minkowski space and in some warped Lorentzian products.
scientific article; zbMATH DE number 1827608