URN 
etd0819108160107 
Author 
ChaoZhong Chen 
Author's Email Address 
m952040011@student.nsysu.edu.tw 
Statistics 
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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 pLaplacian 
Date of Defense 
20080707 
Page Count 
36 
Keyword 
Eigenvalue ratio
SturmLiouville equation

Abstract 
In this thesis, we study the optimal estimate of eigenvalue ratios λ_n/λ_m of the SturmLiouville 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 singlewell 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 onedimensional pLaplacian eigenvalue problem. We show that for the Dirichlet problem of the equation [(y')^(p1)]'=(p1)(λq)y^(p1), where p > 1 and f^(p1)=f^(p1)sgn f =f^(p2)f. The eigenvalue ratios satisfies λ_n/λ_m≤(n/m)^p, assuming that q(x) ≥ 0 and q is a singlewell function on the domain (0, π_p). Again this is an optimal upper estimate. 
Advisory Committee 
WeiCheng Lian  chair
TzonTzer Lu  cochair
FuChuen Chang  cochair
ChunKong Law  advisor

Files 
indicate accessible in a year 
Date of Submission 
20080819 