Potential theory in conical domains
Nicolas Th. Varopoulos · Mathematical Proceedings of the Cambridge Philosophical Society · 1999
We shall denote throughout by Ω⊂≠ℝd 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. Letformula herebe 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 ΩΣformula herewith 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 byformula herethe first exit time from Ω. We shall then define the corresponding heat diffusion kernel and the corresponding ‘probability of life’formula here