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

2009/01/16 by Vladimir Oliker, Oliker, Vladimir
Mathematics · #35J60 #58J05 #Analysis of PDEs (math.AP) #Differential Geometry (math.DG) #FOS: Mathematics #math.AP #math.DG #msc:35J60 #msc:58J05

paper · pdf · doi:10.48550/arxiv.0901.2511

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

arxiv created 2009/09/16 · arxiv updated 2009/12/01

Abstract

For a congruence of straight lines defined by a hypersurface in Rⁿ⁺¹, n ≥ 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 S₁ is a given function.

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