Representing measures on the Royden boundary for solutions of Δu=Pu on a Riemannian manifold

[摘要]

Consider the Royden compactification R* of a Riemannian n-manifold R, Γ = R*\R its Royden boundary, Δ its harmonic boundary and the elliptic differential equation Δu = Pu, P ≥ 0 on R.A regular Borel measure m^P can be constructed on Γ with support equal to the closure of Δ^P = {q ϵ Δ : qhas a neighborhood U in R* with _U^ʃ_ᴖR^{P ˂ ∞} }.Every enegy-finite solution to u (i.e. E(u) = D(u) + ^ʃ_Ru²P ˂ ∞, where D(u) is the Dirichlet integral of u) can be represented by u(z) = ^ʃ_Γu(q)K(z,q)dm^P(q) where K(z,q) is a continuous function on ^Rx Γ .A _P^~_E-function is a nonnegative solution which is the infimum of a downward directed family of energy-finite solutions.A nonzero _P^~_E-function is called _P^~_E-minimal if it is a constant multiple of every nonzero _P^~_E-function dominated by it.THEOREM.There exists a _P^~_E-minimal function if and only if there exists a point in qϵΓsuch that m^P(q) > 0.THEOREM.For q ϵ Δ^P , m^P(q) > 0if and only if m⁰(q) > 0 .

[发布机构] University:California Institute of Technology;Department:Physics, Mathematics and Astronomy

[关键词] Mathematics