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


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.


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 mP can be constructed on Γ with support equal to the closure of ΔP = {q ϵ Δ : q has a neighborhood U in R* with UʃᴖRP ˂ ∞ }. Every enegy-finite solution to u (i.e. E(u) = D(u) + ʃRu2P ˂ ∞, where D(u) is the Dirichlet integral of u) can be represented by u(z) = ʃΓu(q)K(z,q)dmP(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 q ϵ Γ such that mP(q) > 0. THEOREM. For q ϵ ΔP , mP(q) > 0 if and only if m0(q) > 0 .

Item Type:Thesis (Dissertation (Ph.D.))
Subject Keywords:Mathematics
Degree Grantor:California Institute of Technology
Division:Physics, Mathematics and Astronomy
Major Option:Mathematics
Thesis Availability:Public (worldwide access)
Research Advisor(s):
  • Unknown, Unknown
Thesis Committee:
  • Unknown, Unknown
Defense Date:3 April 1970
Record Number:CaltechTHESIS:07302015-141209767
Persistent URL:
Default Usage Policy:No commercial reproduction, distribution, display or performance rights in this work are provided.
ID Code:9069
Deposited On:31 Jul 2015 16:16
Last Modified:25 Jun 2021 21:16

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