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Potential theory in conical domains

Published online by Cambridge University Press:  01 January 1999

N. Th. VAROPOULOS
Affiliation:
Institut Universitaite de France, Université de Paris VI, 4, place Jussieu, Paris

Abstract

We shall denote throughout by Ωd some conical region with vertex at 0. Let Σ=Σd={x∈ℝd; [mid ]x[mid ]=1} denote the unit sphere and by ΩΣ=Σ∩Ω. Let

formula here

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

formula here

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)=rαu(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∈ℝd) 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)∈ℝd the standard Brownian motion in ℝd and by

formula here

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

formula here

Type
Research Article
Copyright
Cambridge Philosophical Society 1999

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