Exit time, Green function and semilinear elliptic equations

Atar, Rami ; Athreya, Siva ; Chen, Zhen-Qing (2009) Exit time, Green function and semilinear elliptic equations Electronic Journal of Probability, 14 . pp. 50-71. ISSN 1083-6489

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Official URL: http://ejp.ejpecp.org/article/view/597

Related URL: http://dx.doi.org/10.1214/EJP.v14-597


Let D be a bounded Lipschitz domain in Rn with n ≥ 2 and τD be the first exit time from D by Brownian motion on Rn. In the first part of this paper, we are concerned with harp estimates on the expected exit time ExD]. We show that if D satisfies a uniform interior cone condition with angle θ ∈ (cos−1(1/√n),π), then c1φ1(x)≤ExD]≤c2φ1(x) on D. Here φ1 is the first positive eigenfunction for the Dirichlet Laplacian on D. The above result is sharp as we show that if Dis a truncated circular cone with angle θ < cos−1(1/√n), then the upper bound for ExD] fails. These results are then used in the second part of this paper to investigate whether positive solutions of the semilinear equation ∆u = up in D, p∈R, that vanish on an open subset Γ⊂∂D decay at the same rate as φ1 on Γ.

Item Type:Article
Source:Copyright of this article belongs to Institute of Mathematical Statistics.
Keywords:Brownian Motion; Exit Time; Feynman-Kac Transform; Lipschitz Dom Ain; Dirichlet Laplacian; Ground State; Boundary Harnack Principle; Green Function Estimates; Semilinear ElLiptic Equation; Schauder’s Fixed Point Theorem
ID Code:99325
Deposited On:28 Mar 2016 10:33
Last Modified:19 May 2016 11:08

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