Skip to main content
We gratefully acknowledge support from the Simons Foundation, member institutions, and all contributors. Donate
arXiv logo
Cornell University Logo

Mathematics > Analysis of PDEs

arXiv:0901.2511 (math)
[Submitted on 16 Jan 2009 (v1), last revised 16 Sep 2009 (this version, v2)]

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

Authors:Vladimir Oliker
View PDF
Abstract: 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.
Comments: 22 pages. This is a substantially revised and expanded version of an earlier posted paper
Subjects: Analysis of PDEs (math.AP); Differential Geometry (math.DG)
MSC classes: 35J60, 58J05
Cite as: arXiv:0901.2511 [math.AP]
  (or arXiv:0901.2511v2 [math.AP] for this version)
  https://doi.org/10.48550/arXiv.0901.2511

Submission history

From: Vladimir Oliker [view email]
[v1] Fri, 16 Jan 2009 16:20:55 UTC (16 KB)
[v2] Wed, 16 Sep 2009 19:03:08 UTC (18 KB)
Full-text links:

Access Paper:

  • View PDF
  • TeX Source
  • Other Formats
view license
Current browse context:
math.AP
< prev   |   next >
new | recent | 2009-01
Change to browse by:
math
math.DG

References & Citations

export BibTeX citation

Bookmark

BibSonomy logo Reddit logo

Bibliographic and Citation Tools

Bibliographic Explorer (What is the Explorer?)
Connected Papers (What is Connected Papers?)
Litmaps (What is Litmaps?)
scite Smart Citations (What are Smart Citations?)

Code, Data and Media Associated with this Article

alphaXiv (What is alphaXiv?)
CatalyzeX Code Finder for Papers (What is CatalyzeX?)
DagsHub (What is DagsHub?)
Gotit.pub (What is GotitPub?)
Hugging Face (What is Huggingface?)
Papers with Code (What is Papers with Code?)
ScienceCast (What is ScienceCast?)

Demos

Replicate (What is Replicate?)
Hugging Face Spaces (What is Spaces?)
TXYZ.AI (What is TXYZ.AI?)

Recommenders and Search Tools

Influence Flower (What are Influence Flowers?)
CORE Recommender (What is CORE?)

arXivLabs: experimental projects with community collaborators

arXivLabs is a framework that allows collaborators to develop and share new arXiv features directly on our website.

Both individuals and organizations that work with arXivLabs have embraced and accepted our values of openness, community, excellence, and user data privacy. arXiv is committed to these values and only works with partners that adhere to them.

Have an idea for a project that will add value for arXiv's community? Learn more about arXivLabs.

Which authors of this paper are endorsers? | Disable MathJax (What is MathJax?)