Results in Mathematics

, 56:519

Kummer Configurations and Sm-Reflector Problems: Hypersurfaces in \({\mathbb{R}}^{n+1}\) with Given Mean Intensity

Article

DOI: 10.1007/s00025-009-0441-6

Cite this article as:
Oliker, V.I. Results. Math. (2009) 56: 519. doi:10.1007/s00025-009-0441-6
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Abstract.

For a congruence of straight lines defined by a hypersurface in \({\mathbb{R}}^{n+1}\), 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 Sm, \(m = 1, 2, \ldots , n\), of principal intensities. The problem of existence and uniqueness of a closed hypersurface with prescribed Sn 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.

Mathematics Subject Classification (2000).

Primary 53C42, 53C40, 35J60 

Keywords.

Reflector problem closed hypersurface with prescribed mean intensity 

Copyright information

© Birkhäuser Verlag Basel/Switzerland 2009

Authors and Affiliations

  1. 1.Department of Mathematics and Computer ScienceEmory UniversityAtlantaUSA

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