Kummer configurations and $S_m-$reflector problems: Hypersurfaces in $\Rn$ with given mean intensity

Details Kummer configurations and $S_m-$reflector problems: Hypersurfaces in $\Rn$ with given mean intensity Kummer configurations and $S_m-$reflector problems: Hypersurfaces in $\Rn$ with given mean intensity

2009-01-16

arxiv.org/abs/0901.2511v2

Mathematics

Analysis of PDEs

22 pages. This is a substantially revised and expanded version of an earlier posted paper

Scientific paper

For a congruence of straight lines defined by a hypersurface in $R^{n+1}, n \geq 1,$ and a field of reflected directions created by a point source we define the notion of intensity in a tangent direction and introduce elementary symmetric functions $S_m, m=1, 2,...,n,$ of {\it principal intensities}. The problem of existence and uniqueness of a closed hypersurface with prescribed $S_n$ is the "reflector problem" extensively studied in recent years. In this paper we formulate and give sufficient conditions for solvability of an analogous problem in which the mean intensity $S_1$ is a given function.

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