发表机构
Tianjin University（天津大学）
机构由 AI 辅助整理，请以论文原文为准。AI 中文总结
本文证明分数阶热方程的最优Li--Yau常数在β趋于2时收敛于经典常数d/2，并给出上下界估计，方法基于Bochner从属公式和Lévy--Itô分解。
AI 中文摘要
对于定义在$\mathbb{R}^d$上、$\beta\in(0,2)$的分数阶热方程$\partial_t u + (-\Delta)^{\beta/2}u=0$，Weber和Zacher在Li--Yau型估计$$(-\Delta)^{\beta/2}\log u(t,\cdot)\leq \frac {C_{\rm LY}(\beta,d)}{t},\quad t>0$$中引入了最优常数$C_{\rm LY}(\beta,d)$。他们提出疑问：当$\beta\uparrow2$时，$C_{\rm LY}(\beta,d)$是否收敛于经典Li--Yau常数$d/2$。在本文中，我们证明答案是肯定的。更精确地，对于每个固定的$d\geq1$，存在常数$K_d>0$，使得$$\frac{d}{\beta}\leq C_{\rm LY}(\beta,d)\leq\frac{d}{\beta}+K_d(2-\beta)\log\frac{e}{2-\beta},\quad 1\leq\beta<2.$$因此，$\lim_{\beta\uparrow2}C_{\rm LY}(\beta,d)=d/2$。证明有两个主要成分。首先，Bochner从属公式结合凹性论证表明，Weber--Zacher公式中$C_{\rm LY}(\beta,d)$的上确界在原点处取得。其次，因子$2-\beta$来源于大距离区域中分数阶热核轮廓的下界，其中Lévy--Itô分解起着重要作用。
英文摘要
For the fractional heat equation $\partial_t u + (-Δ)^{β/2}u=0$ on $\mathbb{R}^d$ with $β\in(0,2)$, Weber and Zacher introduced the optimal constant $C_{\rm LY}(β,d)$ in the Li--Yau type estimate $$(-Δ)^{β/2}\log u(t,\cdot)\leq \frac {C_{\rm LY}(β,d)}{t},\quad t>0.$$ They asked whether $C_{\rm LY}(β,d)$ converges to the classical Li--Yau constant $d/2$ as $β\uparrow2$. In this note we prove that the answer is affirmative. More precisely, for every fixed $d\geq1$ there exists a constant $K_d>0$ such that $$\frac{d}β\leq C_{\rm LY}(β,d)\leq\frac{d}β+K_d(2-β)\log\frac{e}{2-β},\quad 1\leqβ<2.$$ Consequently, $\lim_{β\uparrow2}C_{\rm LY}(β,d)=d/2$. The proof has two main ingredients. First, the Bochner subordination formula, together with a concavity argument, shows that the supremum in Weber--Zacher's formula for $C_{\rm LY}(β,d)$ is attained at the origin. Second, the factor $2-β$ is derived from a lower bound for the fractional heat kernel profile in the large-distance regime, where the Lévy--Itô decomposition plays an important role.
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