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URN etd-0727111-224827
Author Wan-Zhen Wang
Author's Email Address No Public.
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Department Applied Mathematics
Year 2010
Semester 2
Degree Master
Type of Document
Language English
Title p- Laplacian operators with L^1 coefficient functions
Date of Defense 2011-06-10
Page Count 54
Keyword
  • p-Laplacian
  • generalized Prufer substitution
  • Caratheodory problem
  • Sturm oscillation theorem
  • Sturm-Liouville properties
  • Abstract In this thesis, we consider the following one dimensional p-Laplacian eigenvalue problem:
    -((y’/s)^(p-1))’+(p-1)(q-λw)y^(p-1)=0 a.e. on (0,1)   (0.1)
    and satisfy
    αy(0)+ α ’ (y’(0)/s(0))=0 
    βy(1)+β’ (y’(1)/s(1))=0            (0.2)
    where f^(p-1)=|f|^p-2 f=|f|^p-1 sgnf; α, α’, β, β’ ∈R
    such that α^2+α’^2>0 andβ^2+β’^2>0;
    and the functions s,q,w are required to satisfy
    (1) s,q,w∈L^1(0,1);
    (2) for 0≤x≤1, we have s≥0,w≥0 a.e.;
    (3) for any x∈ (0,1), ∫_0^1 s(t)dt>0, ∫_0^x w(t)dt>0,and∫_x^1 w(t)dt>0;
    (4) if for some x_1<x_2,we have∫_ x1^x2 w(t)dt=0,then∫_ x1^x2 |q(t)|dt=0;
    (5) for all n∈N, there is a partition {ζ_i^(n)}_i=1 ^2n of [0,1] such that for any 0<k≤n-1, ∫_ζ_2k^(n)^ ζ_2k+1^(n) w>0 and ∫_ζ_2k+1^(n)^ ζ_2k+2^(n) s>0.
    We call the above conditions Atkinson conditions, first introduce in [1].There conditions include the case when s,q,w∈L^1(0,1) and s,w>0 a.e.
    We use a generalized Prufer substitution and Caratheodory theorem to prove the existence and uniqueness for the solution of the initial value problem of (0.1) above. Then we generalize the Sturm oscillation theorem to one dimensional p-Laplacian and establish the Sturm-Liouville properties of the p-Laplacian operators with L^1 coefficient functions. Our results filled up some gaps in Binding-Drabek [3].
    Advisory Committee
  • W.C. Lian - chair
  • Tzon-Tzer Lu - co-chair
  • Tsung-Lin Lee - co-chair
  • Chun-Kong Law - advisor
  • Files
  • etd-0727111-224827.pdf
  • indicate accessible in a year
    Date of Submission 2011-07-27

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