### Title page for etd-0819108-160107

URN etd-0819108-160107 Chao-Zhong Chen m952040011@student.nsysu.edu.tw This thesis had been viewed 5178 times. Download 1242 times. Applied Mathematics 2007 2 Master English Optimal upper bounds of eigenvalue ratios for the p-Laplacian 2008-07-07 36 Eigenvalue ratio Sturm-Liouville equation In this thesis, we study the optimal estimate of eigenvalue ratios λ_n/λ_m of theSturm-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 thatHorvath and Kiss [10] used. Furthermore our approach can be generalized to studythe one-dimensional p-Laplacian eigenvalue problem. We show that for the Dirichletproblem 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. Wei-Cheng Lian - chair Tzon-Tzer Lu - co-chair Fu-Chuen Chang - co-chair Chun-Kong Law - advisor indicate accessible in a year 2008-08-19

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