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JOURNALS // Matematicheskii Sbornik // Archive

Mat. Sb., 1992 Volume 183, Number 7, Pages 49–64 (Mi sm1056)

This article is cited in 13 papers

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

A. A. Klyachin, V. M. Miklyukov


Abstract: 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.

UDC: 514

MSC: 49Q20, 53B30

Received: 10.07.1991


 English version:
Russian Academy of Sciences. Sbornik. Mathematics, 1993, 76:2, 305–316

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