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Let Ξ(x) be the distribution of convex sets over a domain D ⊂ R and let φ : ∂D → R be a function. We consider the existence problem of locally Lipschitz functions f defined in the domain D so that f |∂D = φ and ∇f(x) ∈ Ξ(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.

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