Potential theory in conical domains

Abstract

We shall denote throughout by Ω^⊂_≠ℝ^d some conical region with vertex at 0. Let Σ=Σ_d={x∈ℝ^d; [mid ]x[mid ]=1} denote the unit sphere and by Ω_Σ=Σ∩Ω. Let

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be the expression of the Laplacian in polar coordinates x=(r, σ) (r=[mid ]x[mid ], σ∈Σ) where Δ_r denote the corresponding r-dimensional spherical Laplacian. For d[ges ]2 we can always solve the first eigenfunction problem for Ω_Σ

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with Dirichlet boundary conditions. We shall assume throughout that ∂Ω is sufficiently regular to ensure that u is continuous up to the boundary and vanishes there. With α>0α(α+d−2)=λ the function u(x)=r^αu(σ)= [mid ]x[mid ]^αu(x/[mid ]x[mid ]) is then harmonic in Ω. It should be observed that α[ges ]1 if Ω is convex. If we further assume, as we shall do in this paper, that ∂Ω is Lipschitz, u(x) (x∈ℝ^d) is the unique, up tp multiplicative constant positive harmonic function in Ω that vanishes at the boundary ∂Ω This function is homogeneous and is called the réduite of Ω (cf. [1]). We shall denote throughout by α=degΩ the homogenity degree of u. If d=1 the réduite is u(x)≡Cx(x>0).

We shall denote by b(t)∈ℝ^d the standard Brownian motion in ℝ^d and by

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the first exit time from Ω. We shall then define the corresponding heat diffusion kernel and the corresponding ‘probability of life’

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