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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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