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Information Center for Mathematical Science

논문검색

Information Center for Mathematical Science

논문검색

Glasnik Matematicki Serja III.
( Vol. 39 NO.2 / (2004))
Estimates on the Dirichlet heat kernel of domains above the graphs of bounded ${\sl C}^{1.1}$ fuctions
Renming Song,
Pages. 273-286
Abstract Suppose that ${sl D}$ is the domain in $mathbb{R}^{it d}, {it d} geq 3$, above the graph of a bounded ${sl C}^{1.1}$ function $Gamma : mathbb{R}^{{it d}-1} o mathbb{R}$ and that ${it p}^{sl D}({it t,x,y})$ is the Dirichlet heat kernel in ${sl D}$. In this paper we show that there exist positive constants ${sl C}_1, {sl C}_2, {sl C}_3$ and ${sl C}_4$ such that for all ${it t} > 0$ and ${it x, y} in {sl D}$, $${sl C}_1(frac{{sf p}({it x}){sf p}({it y}){{it t}} wedge 1){it t}^{-{it d}/2}{sf e}^{-frac{{sl C}_2|{it x-y}|^2}{{it t}} leq {it p}^{sl D}({it t,x,y}),//{it p}^{sl D}({it t,x,y}) leq {sl C}_3(frac{{sf p}({it x}){sf p}({it y}){{it t}} wedge 1){it t}^{-{it d}/2}{sf e}^{-frac{{sl C}_4|{it x-y}|^2}{{it t}}$$, where ${sf p}({it x})$ stands for the distance between ${it x}$ and $partial{sl D}$.
Contents 1.Introduction
2.Preliminaries
3.The main results
4.Applications to subordinate killed Brownian motion
Key words Dirichlet heat kernels, Green fucntions, killed Brownian motions
Mathmatical Subject Classification 35K05, 60J60, 60J45, 60J75