| Título: | Exit Time, Green Function and Semilinear Elliptic Equations |
| Autores: |
Atar, Rami; Technion Athreya, Siva; Indian Statistical Institute Chen, Zhen-Qing; University of Washington |
| Fecha: | 2009-01-01 |
| Publicador: | Electronic journal of probability |
| Fuente: |
 |
| Tipo: |
Peer-reviewed Article |
| Tema: |
Brownian motion; exit time; Feynman-Kac transform; Lipschitz domain; Dirichlet Laplacian; ground state; boundary Harnack principle; Green function estimates; semilinear elliptic equation; Schauder's fixed point theorem 60H30; 60J45; 35J65; 60J35; 35J10 |
| Descripción: | Let $D$ be a bounded Lipschitz domain in $R^n$ with $n\geq 2$ and $\tau_D$ be the first exit time from $D$ by Brownian motion on $R^n$. In the first part of this paper, we are concerned with sharp estimates on the expected exit time $E_x [ \tau_D]$. We show that if $D$ satisfies a uniform interior cone condition with angle $\theta \in ( \cos^{-1}(1/\sqrt{n}), \pi )$, then $c_1 \varphi_1(x) \leq E_x [\tau_D] \leq c_2 \varphi_1 (x)$ on $D$. Here $\varphi_1$ is the first positive eigenfunction for the Dirichlet Laplacian on $D$. The above result is sharp as we show that if $D$ is a truncated circular cone with angle $\theta < \cos^{-1}(1/\sqrt{n})$, then the upper bound for $E_x [\tau_D]$ fails. These results are then used in the second part of this paper to investigate whether positive solutions of the semilinear equation $\Delta u = u^{p}$ in $ D,$ $p\in R$, that vanish on an open subset $\Gamma \subset \partial D$ decay at the same rate as $\varphi_1$ on $\Gamma$. |
| Idioma: | No aplica |
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