2018/09/27 by David Jerison, Jerison, David
Mathematics · #35A15 #35B35 #Analysis of PDEs (math.AP) #FOS: Mathematics #math.AP #msc:35A15 #msc:35B35
paper · pdf · doi:10.48550/arxiv.1809.10759
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
arxiv created 2019/02/01 · arxiv updated 2019/02/04
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, Poincaré 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.