AI 中文总结
本文定义容度距离，证明其对应的哈代不等式成立，正面回答了Maz'ya的问题8，且该不等式对参数α的依赖是最优的，证明结合了被杀布朗运动的变时半群估计与容度导出的有限时间退出估计。
AI 中文摘要
设$n\ge3$，$\Omega$是$\mathbb R^n$中的开集，$F:=\mathbb R^n\setminus\Omega$，$\alpha\in(0,\infty)$。对任意$x\in\Omega$，定义容度距离$d_\alpha(x):= \inf\left\{ r>0: \operatorname{cap}(\overline{F\cap B(x,r)}) \ge \alpha\operatorname{cap}(B(\mathbf0,r)) \right\}$。本文证明存在仅依赖$n$的正常数$C_n$，使得对任意$\alpha\in(0,1]$和任意$u\in C_{\rm{c}}^\infty(\Omega)$，有$\int_\Omega \frac{|u(x)|^2}{d_\alpha(x)^2}\\,d x \le \frac{C_n}{\alpha^{2}} \int_\Omega|\nabla u(x)|^2\\,d x$，这正面回答了Mazya文献[25]中的问题8。此外，该对$\alpha$的依赖是最优的：存在仅依赖$n$的正常数$c_n$，使得对每个$\alpha\in(0,1]$，可构造有界连通域$\Omega_\alpha$，其上上述哈代不等式的最优常数至少为$\frac{c_n}{\alpha^{2}}$。证明结合了被杀布朗运动的变时半群估计与由容度导出的有限时间退出估计。
英文摘要
We study Brownian motion killed upon exiting arbitrary open sets $Ω\subset\mathbb R^n$, $n\in[3,\infty)\cap\mathbb N$. Let $α\in(0,1]$. The infimal radius $d_α(x)$ at which the absorber near $x$ carries an $α$-fraction of the Newtonian capacity of the ball of the same radius provides a local measure of electrostatic trapping. We prove that this capacitary loading forces an absorption probability of at least $c_nα$ within a dimension-dependent multiple of the diffusive time scale $d_α(x)^2$. Independently, if $p\in (0,1)$ and $τ_p(x)$ is the time by which Brownian motion started at $x$ has been absorbed with probability at least $p$, then \begin{align*} \int_Ω\frac{|u\left(x\right)|^2}{τ_p\left(x\right)}\,\mathrm d x \le\frac{2}{p^2}\int_Ω|\nabla u\left(x\right)|^2\,\mathrm d x \end{align*} for every $u\in C_{\mathrm{c}}^\infty(Ω)$. Together these statements give a capacitary Hardy inequality with constant $C_nα^{-2}$ on every open set, answering a problem of Maz'ya, where $C_n$ is a positive constant and, moreover, the exponent of $α^{-2}$ is sharp. As applications, we derive corresponding estimates for survival probabilities, a corresponding spectral lower bound (including a recovery of the Maz'ya--Shubin lower bound), heat dissipation, Schrödinger forms, and Fisher information.
Comments28 pages
