In this thesis, we study the optimal estimate of eigenvalue ratios £f_n/£f_m of the
Sturm-Liouville equation with Dirichlet boundary conditions on (0, £k). In 2005, Horvath and Kiss [10] showed that £f_n/£f_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 £f_n/£f_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(£s£c(x)), y'(x)=r(x)£scos(£s£c(x)), where £s =
¡Ô£f. This modified phase seems to be more effective than the phases £p and £r 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)(£f-q)y^(p-1), where p > 1 and f^(p-1)=|f|^(p-1)sgn f =|f|^(p-2)f. The eigenvalue ratios satisfies £f_n/£f_m≤(n/m)^p, assuming that q(x) ≥ 0 and q is a single-well function on the domain (0, £k_p). Again this is an optimal upper estimate.
| Identifer | oai:union.ndltd.org:NSYSU/oai:NSYSU:etd-0819108-160107 |
| Date | 19 August 2008 |
| Creators | Chen, Chao-Zhong |
| Contributors | Chun-Kong Law, Tzon-Tzer Lu, Fu-Chuen Chang, Wei-Cheng Lian |
| Publisher | NSYSU |
| Source Sets | NSYSU Electronic Thesis and Dissertation Archive |
| Language | English |
| Detected Language | English |
| Type | text |
| Format | application/pdf |
| Source | http://etd.lib.nsysu.edu.tw/ETD-db/ETD-search/view_etd?URN=etd-0819108-160107 |
| Rights | withheld, Copyright information available at source archive |
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