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The Two Hyperplane Conjecture
- Source :
- arXiv
- Publication Year :
- 2018
-
Abstract
- We introduce a conjecture that we call the {\it Two Hyperplane Conjecture}, saying that an isoperimetric surface that divides a convex body in half by volume is trapped between parallel hyperplanes. The conjecture is motivated by an approach we propose to the {\it Hots Spots Conjecture} of J. Rauch using deformation and Lipschitz bounds for level sets of eigenfunctions. We will relate this approach to quantitative connectivity properties of level sets of solutions to elliptic variational problems, including isoperimetric inequalities, Poincar\'e inequalities, Harnack inequalities, and NTA (non-tangentially accessibility). This paper mostly asks questions rather than answering them, while recasting known results in a new light. Its main theme is that the level sets of least energy solutions to scalar variational problems should be as simple as possible.<br />Comment: 22 pages, this new version corrects one word in the introduction, Aguilera is the name of the first author of a paper cited (not Athanosopoulos). Thanks to Joel Spruck for pointing out this error
- Subjects :
- Pure mathematics
Conjecture
35B35, 35A15
Applied Mathematics
General Mathematics
010102 general mathematics
Scalar (mathematics)
Eigenfunction
01 natural sciences
symbols.namesake
Mathematics - Analysis of PDEs
Hyperplane
0103 physical sciences
Poincaré conjecture
FOS: Mathematics
symbols
Convex body
010307 mathematical physics
0101 mathematics
Isoperimetric inequality
Analysis of PDEs (math.AP)
Mathematics
Subjects
Details
- Language :
- English
- Database :
- OpenAIRE
- Journal :
- arXiv
- Accession number :
- edsair.doi.dedup.....c7e1eda9d349c4fd716eb899ec3b5d22