Exit time, green function and semilinear elliptic equations

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 sharp 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 ≤ 𝔼_x[τ^D] ≤ c₁φ₁ 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 D is 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 Δu = u^p p ∈ R, that vanish on an open subset Γ ⊂ ∂D decay at the same rate as φ₁ on Γ.