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arXiv 1102.1454math.PRmath.CA
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类半空间域上$Δ+Δ^{α/2}$的整体热核估计

Global Heat Kernel Estimates for $Δ+Δ^{α/2}$ in Half-space-like domains

Zhen-Qing Chen, Panki Kim, Renming Song

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摘要：

假设$d\ge 1$和$α\in (0, 2)$ .本文利用概率方法，建立了$t>0$中${\mathbb R}^d$中半空间类$C^{1, 1}$域上$\{Δ+ a^αΔ^{α/2}; \ a\in (0, 1]\}$的Dirichlet热核的精确双边逐点估计.半空间域的大时间估计与有界域的大时间估计有很大的不同。我们的估计在$a \in (0, 1]$中是一致的，因为估计中的常数与$a\in (0, 1]$无关。因此，利用$a\to 0$得到了半空间区域上布朗运动的Dirichlet热核估计.在时间$t$上积分热核估计，得到了${\mathbb R}^d$中类半空间$C^{1, 1}$域上$\{Δ+ a^αΔ^{α/2}; \ a\in (0, 1]\}$的格林函数的一致尖锐双边估计.

英文摘要：

Suppose that $d\ge 1$ and $α\in (0, 2)$. In this paper, by using probabilistic methods, we establish sharp two-sided pointwise estimates for the Dirichlet heat kernels of $\{Δ+ a^αΔ^{α/2}; \ a\in (0, 1]\}$ on half-space-like $C^{1, 1}$ domains in ${\mathbb R}^d$ for all time $t>0$. The large time estimates for half-space-like domains are very different from those for bounded domains. Our estimates are uniform in $a \in (0, 1]$ in the sense that the constants in the estimates are independent of $a\in (0, 1]$. Thus it yields the Dirichlet heat kernel estimates for Brownian motion in half-space-like domains by taking $a\to 0$. Integrating the heat kernel estimates in time $t$, we obtain uniform sharp two-sided estimates for the Green functions of $\{Δ+ a^αΔ^{α/2}; \ a\in (0, 1]\}$ in half-space-like $C^{1, 1}$ domains in ${\mathbb R}^d$.

机构、总结、中文摘要均由AI分析而来，可能存在错误，请以论文正文为准。

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