Citation
Chow, Kwangnan (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/D80CCD98. https://resolver.caltech.edu/CaltechTHESIS:07302015141209767
Abstract
Consider the Royden compactification R* of a Riemannian nmanifold 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 enegyfinite solution to u (i.e. E(u) = D(u) + ^{ʃ}_{R}u^{2}P ˂ ∞, where D(u) is the Dirichlet integral of u) can be represented by u(z) = ^{ʃ}_{Γ}u(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 energyfinite 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 m^{P}(q) > 0. THEOREM. For q ϵ Δ^{P} , m^{P}(q) > 0 if and only if m^{0}(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): 

Thesis Committee: 

Defense Date:  3 April 1970 
Record Number:  CaltechTHESIS:07302015141209767 
Persistent URL:  https://resolver.caltech.edu/CaltechTHESIS:07302015141209767 
DOI:  10.7907/D80CCD98 
Default Usage Policy:  No commercial reproduction, distribution, display or performance rights in this work are provided. 
ID Code:  9069 
Collection:  CaltechTHESIS 
Deposited By:  INVALID USER 
Deposited On:  31 Jul 2015 16:16 
Last Modified:  29 May 2024 18:16 
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