标题： 热斑猜想失败的精确界限 显示英文标题

标题： Sharp bounds on the failure of the hot spots conjecture

摘要： 区域$\Omega\subset \mathbb{R}^d$的热点比率衡量了在该区域上 Rauch 的热点猜想失败的程度。 我们确定了所有连通的 Lipshitz 区域$\Omega\subset \mathbb{R}^d$上该比率的最大可能值，对于任何维度$d$。 当$d\to \infty$时，我们证明这个最大比率收敛到$\sqrt{e}$，这在渐近意义上与 Mariano、Panzo 和 Wang 之前最好的已知上限相匹配。 对于$d\ge 2$，我们证明极值化热点比率的集合不存在，极值化序列必须以定量速率收敛到一个球体。 然后我们给出了第一 Neumann 特征函数超过其最大边界值的集合的测度的精确界限。 由此我们推导出，当$d\to \infty$时，热点猜想在“测度”意义上渐近成立。 显示英文摘要

摘要： The hot spots ratio of a domain $\Omega\subset \mathbb{R}^d$ measures the degree of failure of Rauch's hot spots conjecture on that domain. We identify the largest possible value of this ratio over all connected Lipschitz domains $\Omega\subset \mathbb{R}^d$, for any dimension $d$. As $d\to \infty$, we show that this maximal ratio converges to $\sqrt{e}$, which asymptotically matches the previous best known upper bound by Mariano, Panzo and Wang. For $d\ge 2$, we show that sets extremizing the hot spots ratio do not exist, and extremizing sequences must converge to a ball at a quantitative rate. We then give a sharp bound on the measure of the set for which the first Neumann eigenfunction exceeds its maximal boundary value. From this we deduce that the hot spots conjecture is asymptotically true "in measure'' as $d\to \infty$.