Exit time, Green function and semilinear elliptic equations

Abstract

Let D be a bounded Lipschitz domain in Rⁿ with n ≥ 2 and τ_D 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 E_x[τ_D]. We show that if D satisfies a uniform interior cone condition with angle θ ∈ (cos⁻¹(1/√n),π), then c₁φ₁(x)≤E_x[τ_D]≤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 E_x[τ_D] fails. These results are then used in the second part of this paper to investigate whether positive solutions of the semilinear equation ∆u = u^p in D, p∈R, that vanish on an open subset Γ⊂∂D decay at the same rate as φ₁ on Γ.