A Survey on Extremal Problems of Eigenvalues

Abstract

Given an integrable potential $q\in L^1([0,1],ℝ)$, 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; ${\parallel q\parallel }_{L^1}=r$. Note that the $L^1$ spheres and $L^1$ balls are nonsmooth, noncompact domains of the Lebesgue space $(L^1{([0,1],ℝ),\parallel ·\parallel }_{L^1})$. 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 $L^{\alpha }([0,1],ℝ)$, 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.