Abstract
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 $\Gamma$.
Highlights
Let n ≥ 2 and X a Brownian motion on n
We show in this paper that, (1.2) holds on any bounded Lipschitz domain D with Lipschitz constant strictly less than 1/ n − 1, and fails on some Lipschitz domain with Lipschitz constant strictly larger than 1/ n − 1
(iii) While the lower bound for x [τD] in (1.4) holds for every bounded Lipschitz domain, we show in Theorem 3.3 that the condition on θ is sharp for the upper bound ( for (1.2))
Summary
Let n ≥ 2 and X a Brownian motion on n. The result below states that (1.2) holds for bounded Lipschitz domains in n satisfying the interior cone condition with common angle strictly larger than cos−1(1/ n). Let D be a bounded Lipschitz domain in n with n ≥ 2 satisfying the interior cone condition with common angle θ ∈ cos−1(1/ n), π. Under certain regularity conditions on D ⊂ n and φ, where n ≥ 3, the existence of solutions to (1.9), bounded below by a positive harmonic function, was established in [6] when f satisfies the condition that −u ≤ f (u) ≤ u for |u| < ǫ for some ǫ > 0, and in [1] the case when 0 ≤ f (u) ≤ u−α for some α ∈ (0, 1) was resolved.
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