Geodesic
Traces of functions with spacelike graphs, and the extension problem under restrictions on the gradient
Sbornik. Mathematics, Tome 76 (1993) no. 2, pp. 305-316 Cet article a éte moissonné depuis la source Math-Net.Ru

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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. 
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A. A. Klyachin; V. M. Miklyukov. Traces of functions with spacelike graphs, and the extension problem under restrictions on the gradient. Sbornik. Mathematics, Tome 76 (1993) no. 2, pp. 305-316. http://geodesic.mathdoc.fr/item/SM_1993_76_2_a3/

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