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Abstract and Applied Analysis
Volume 2012 (2012), Article ID 670463, 26 pages
http://dx.doi.org/10.1155/2012/670463
Review Article

A Survey on Extremal Problems of Eigenvalues

1Department of Mathematical Sciences, Tsinghua University, Beijing 100084, China
2Zhou Pei-Yuan Center for Applied Mathematics, Tsinghua University, Beijing 100084, China

Received 3 July 2012; Accepted 7 August 2012

Academic Editor: Yeong-Cheng Liou

Copyright © 2012 Ping Yan and Meirong Zhang. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.

Abstract

Given an integrable potential , the Dirichlet and the Neumann eigenvalues and 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 metric for q is given; . Note that the spheres and balls are nonsmooth, noncompact domains of the Lebesgue space . 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 , will be used. Then the problems will be solved by passing . 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.