Title page for etd-0819108-160107


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URN etd-0819108-160107
Author Chao-Zhong Chen
Author's Email Address m952040011@student.nsysu.edu.tw
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Department Applied Mathematics
Year 2007
Semester 2
Degree Master
Type of Document
Language English
Title Optimal upper bounds of eigenvalue ratios for the p-Laplacian
Date of Defense 2008-07-07
Page Count 36
Keyword
  • Eigenvalue ratio
  • Sturm-Liouville equation
  • Abstract In this thesis, we study the optimal estimate of eigenvalue ratios λ_n/λ_m of the
    Sturm-Liouville equation with Dirichlet boundary conditions on (0, π). In 2005, Horvath and Kiss [10] showed that λ_n/λ_m≤(n/m)^2 when the potential function q ≥ 0 and is a single-well function. Also this is an optimal upper estimate, for equality holds if and only if q = 0. Their result gives a positive answer to a problem posed by Ashbaugh and Benguria [2], who earlier showed that λ_n/λ_1≤n^2 when q ≥ 0.
     Here we first simplify the proof of Horvath and Kiss [10]. We use a modified Prufer substitutiony(x)=r(x)sin(ωθ(x)), y'(x)=r(x)ωcos(ωθ(x)), where ω =
    √λ. This modified phase seems to be more effective than the phases φ and ψ that
    Horvath and Kiss [10] used. Furthermore our approach can be generalized to study
    the one-dimensional p-Laplacian eigenvalue problem. We show that for the Dirichlet
    problem of the equation -[(y')^(p-1)]'=(p-1)(λ-q)y^(p-1), where p > 1 and f^(p-1)=|f|^(p-1)sgn f =|f|^(p-2)f. The eigenvalue ratios satisfies λ_n/λ_m≤(n/m)^p, assuming that q(x) ≥ 0 and q is a single-well function on the domain (0, π_p). Again this is an optimal upper estimate.
    Advisory Committee
  • Wei-Cheng Lian - chair
  • Tzon-Tzer Lu - co-chair
  • Fu-Chuen Chang - co-chair
  • Chun-Kong Law - advisor
  • Files
  • etd-0819108-160107.pdf
  • indicate accessible in a year
    Date of Submission 2008-08-19

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