A Survey on Extremal Problems of Eigenvalues

Abstract

Given an integrable potentialq∈L1([0,1],ℝ), the Dirichlet and the Neumann eigenvaluesλnD(q)andλnN(q)of the Sturm-Liouville operator with the potentialqare defined in an implicit way. In recent years, the authors and their collaborators have solved some basic extremal problems concerning these eigenvalues when theL1metric forqis given;∥q∥L1=r. Note that theL1spheres andL1balls are nonsmooth, noncompact domains of the Lebesgue space(L1([0,1],ℝ),∥·∥L1). 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 spacesLα([0,1],ℝ),1<α<∞will be used. Then theL1problems will be solved by passingα↓1. Corresponding extremal problems for eigenvalues of the one-dimensionalp-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.