Traces of functions with spacelike graphs, and the extension problem under restrictions on the gradient

Let $D\subset\mathbb{R}^n$ be a domain, and suppose that for each $x\in D$ a subset $\Xi(x)$ of $\mathbb{R}^n$ is given. The problem is posed of finding conditions under which a function $\varphi(x)$ defined on the boundary $\partial D$ can be extended to a $C^1$-function $f(x)$ defined in $D$ and such that the gradient satisfies $\nabla f(x)\in\Xi(x)$.
This problem is solved for the case when $\Xi(x)$ is a continuous distribution of bounded convex sets. An application is given to the description of the trace of a function with spacelike graph in a Lorentzian warped product.