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URN etd-0531110-195414
Author Wei-Chuan Wang
Author's Email Address No Public.
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Department Applied Mathematics
Year 2009
Semester 2
Degree Ph.D.
Type of Document
Language English
Title Direct and inverse problems for one-dimensional p-Laplacian operators
Date of Defense 2010-05-28
Page Count 59
Keyword
  • p-Laplacian
  • inverse problems
  • Abstract In this thesis, direct and inverse problems concerning nodal solutions associated with the one-dimensional p-Laplacian operators are studied. We first consider the eigenvalue
    problem on (0, 1),
    −(y0(p−1))0 + (p − 1)q(x)y(p−1) = (p − 1) λw(x)y(p−1) (0.1)
    Here f(p−1) := |f|p−2f = |f|p−1 sgn f. This problem, though nonlinear and degenerate, behaves very similar to the classical Sturm-Liouville problem, which is the special case
    p = 2. The spectrum {λk} of the problem coupled with linear separated boundary conditions are discrete and the eigenfunction yn corresponding toλn has exactly n−1 zeros in (0, 1). Using a Pr‥ufer-type substitution and properties of the generalized sine function, Sp(x), we solve the reconstruction and stablity issues of the inverse nodal problems for Dirichlet boundary conditions, as well as periodic/antiperiodic boundary conditions whenever w(x) λ 1. Corresponding Ambarzumyan problems are also solved.
    We also study an associated boundary value problem with a nonlinear nonhomogeneous
    term (p−1)w(x) f(y(x)) on the right hand side of (0.1), where w is continuously differentiable and positive, q is continuously differentiable and f is positive and Lipschitz
    continuous on R+, and odd on R such that
    f0 := lim
    y!0+
    f(y)
    yp−1 , f1 := lim
    y!1
    f(y)
    yp−1 .
    are not equal. We extend Kong’s results for p = 2 to general p > 1, which states that whenever an eigenvalue _n 2 (f0, f1) or (f1, f0), there exists a nodal solution un
    having exactly n − 1 zeros in (0, 1), for the above nonhomogeneous equation equipped
    with any linear separated boundary conditions.
    Although it is known that there are indeed some differences, Our results show that the one-dimensional p-Laplacian operator is still very similar to the Sturm-Liouville operator, in aspects involving Pr‥ufer substitution techniques.
    Advisory Committee
  • Chao-Nien Chen - chair
  • Wei-Cheng Lian - co-chair
  • Chun-Kong Law - advisor
  • Files
  • etd-0531110-195414.pdf
  • indicate accessible in a year
    Date of Submission 2010-05-31

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