The Two Hyperplane Conjecture
The Two Hyperplane Conjecture
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两个超平面猜想
DOI：
10.1007/s10114-019-8241-8
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发表时间：
2018
期刊：
影响因子：
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通讯作者：
D. Jerison
中科院分区：
文献类型：
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作者：
D. Jerison
我们引入一个猜想，称为两个超平面猜想，即按体积将凸体分成两半的等周曲面被困在平行超平面之间。该猜想的动机是我们针对 J. Rauch 的热点猜想提出的一种方法，该方法使用特征函数水平集的变形和 hipschitz 界限。我们将把这种方法与椭圆变分问题解水平集的定量连通性联系起来，包括等周不等式、庞加莱不等式、哈纳克不等式和 NTA（非切向可达性）。本文主要提出问题而不是回答问题，同时以新的眼光重新审视已知的结果。其主题是标量变分问题的最小能量解的水平集应该尽可能简单。
We introduce a conjecture that we call the Two Hyperplane Conjecture, saying that an isoperimetric surface that divides a convex body in half by volume is trapped between parallel hy-perplanes. The conjecture is motivated by an approach we propose to the Hots Spots Conjecture of J. Rauch using deformation and hipschitz 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, Poincare 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.