A survey on extremal problems of eigenvalues
type: Item
instance of: scholarly article
title: A survey on extremal problems of eigenvalues
zbMATH Open document ID: 1296.34172
DOI: 10.1155/2012/670463
author: Ping Yan
author: Meirong Zhang
published in: Abstract and Applied Analysis
publication date: 2012-12-18T00:00:00Z
review text: Given an integrable potential \(q\in L^1 ([0,1],\mathbb{R})\), the Dirichlet and the Neumann eigenvalues \(\lambda_n^D (q)\) and \(\lambda_n^N (q)\) of the Sturm-Liouville operator with the potential \(q\) are defined in an implicit way. In recent years, the authors and their collaborators have solved some basic extremal problems concerning these eigenvalues when the \(L^1\) metric for \(q\) is given; \(\|q\|_{L^1} =r\). Note that the \(L^1\) spheres and \(L^1\) balls are nonsmooth, noncompact domains of the Lebesgue space \(L^1 ([0,1],\mathbb{R})\). To solve these extremal problems, we will reveal some deep results on the dependence of eigenvalues on potentials. Moreover, the variational method for the approximating extremal problems on the balls of the spaces \(L^{\alpha} ([0,1],\mathbb{R})\), \(1<\alpha<\infty\), will be used. Then the \(L^1\) problems will be solved by passing \(\alpha\downarrow 1\). Corresponding extremal problems for eigenvalues of the one-dimensional \(p\)-Laplacian with integrable potentials have also been solved. The results can yield optimal lower and upper bounds for these eigenvalues. This paper will review the most important ideas and techniques in solving these difficult and interesting extremal problems. Some open problems will also be imposed.
Mathematics Subject Classification ID: 34L15
Mathematics Subject Classification ID: 34B24
Mathematics Subject Classification ID: 47A75
zbMATH DE Number: 6116429
zbMATH Keywords: extremal problems
zbMATH Keywords: eigenvalues
zbMATH Keywords: Dirichlet eigenvalues
zbMATH Keywords: Neumann eigenvalues
Wikidata QID: Q58695832
MaRDI profile type: Publication
full work available at URL: https://doi.org/10.1155/2012/670463
OpenAlex ID: W2159798302
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description: scientific article; zbMATH DE number 6116429