A Survey on Extremal Problems of Eigenvalues

and Applied Analysis 3 The final solution to problems in 1.3 can yield optimal lower and upper bounds of eigenvalues. Actually, from 1.3 , we have In ∥q ∥ ∥ 1 ) ≤ λn ( q ) ≤ Mn ∥q ∥ ∥ 1 ) ∀q ∈ L1, 1.4 which are optimal. Several extremal problems on Dirichlet, Neumann, and periodic eigenvalues of the Sturm-Liouville operator and the p-Laplace operator have been studied recently, where the potentials are confined to different sets such as balls or spheres in L1. For definition of these problems, see 4.1 . In the following sections, we give a slightly detailed description on these results. Some topological facts about L, α ≥ 1, are listed in Section 2. The essential of this section is that those “bad” balls or spheres in L1 neither smooth nor weak compact can be approximated by “good” balls or spheres in L, α > 1 smooth and weak compact . Section 3 is devoted to some properties of eigenvalues, including the scaling results which enable us to consider only those integrable potentials defined on the interval 0, 1 , the relationship between the first and higher-order eigenvalues which plays a role to enable us to consider only the first order, and the strong continuity in weak topology together with the continuous differentiability in strong topology which enable us to apply the variational method to problems confined on those “good” balls or spheres in L, α > 1. In Section 4 we introduce how variational method is applied to get critical equations for the extremal problems for cases α > 1. After analysis on the critical equations, these extremal values are determined by some singular integrals. However, they cannot be expressed by elementary functions. Section 5 deals with the extremal problems in L1 spaces via limiting approaches. Final results of extremal values in L1 case are stated in this section. For the Sturm-Liouville operator, these extremal values can be expressed by elementary functions of the radius r. In Section 6, the corresponding extremal problems for eigenvalues of measure differential equations 26, 27 are discussed briefly. The minimizing measures for the problems in 1.3 are explained. In Section 7, two open problems for further study are imposed. One is on the eigenvalue gaps and the other is on the corresponding extremal problems of eigenvalues of the beam equation with integrable potentials. 2. Some Topological Facts on L Spaces In the Lebesgue space L, 1 ≤ α ≤ ∞, the usual L topology is induced by L norm ‖ · ‖α. Besides this strong topology, one has also the weak topology wα which is defined as follows 24, 25 . Definition 2.1. Let qn, q ∈ L. We say that qn is weakly convergent to q, written as qn wα −−→ q in L, or qn → q in L,wα , if