Representing Measures on the Royden Boundary for Solutions of Δu=Pu on a Riemannian Manifold

• Main Author: Chow, Kwang-nan
• Format: Others
• Language: en
• Published: 1970
• Online Access: https://thesis.library.caltech.edu/9069/1/Chow_kn_1970.pdf; Chow, Kwang-nan (1970) Representing Measures on the Royden Boundary for Solutions of Δu=Pu on a Riemannian Manifold. Dissertation (Ph.D.), California Institute of Technology. doi:10.7907/D80C-CD98. https://resolver.caltech.edu/CaltechTHESIS:07302015-141209767 <https://resolver.caltech.edu/CaltechTHESIS:07302015-141209767>

<p>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 : q has 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. <u>THEOREM</u>. There exists a _P^~_E-minimal function if and only if there exists a point in q ϵ Γ such that m^P(q) > 0. <u>THEOREM</u>. For q ϵ Δ^P , m^P(q) > 0 if and only if m⁰(q) > 0 .</p>