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Exit time, Green function and semilinear elliptic equations. (English) Zbl 1190.60056
Summary: Let $$D$$ be a bounded Lipschitz domain in $$\mathbb R^n$$ with $$n\geq 2$$ and $$\tau_D$$ be the first exit time from $$D$$ by Brownian motion on $$\mathbb R^n$$. 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 $$c_1 \varphi_1(x)\leq\mathbb E_x [\tau_D]\leq c_2\varphi_1(x)$$ 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 $$\mathbb E_x [\tau_D]$$ 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 $$D$$, $$p\in\mathbb R$$, that vanish on an open subset $$\Gamma\subset\partial D$$ decay at the same rate as $$\varphi_1$$ on $$\Gamma$$.

MSC:
 60H30 Applications of stochastic analysis (to PDEs, etc.) 60J45 Probabilistic potential theory 35J65 Nonlinear boundary value problems for linear elliptic equations
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