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- W2894097775 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." @default.
- W2894097775 created "2018-10-05" @default.
- W2894097775 creator A5076249270 @default.
- W2894097775 date "2018-09-27" @default.
- W2894097775 modified "2023-09-27" @default.
- W2894097775 title "The Two Hyperplane Conjecture." @default.
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