A variational approach to the optimal locations of the nodes of the second Dirichlet eigenfunctions

Abstract

By a node of a Sturm-Liouville problem, it means an interior zero of an eigenfunction. In this paper, by considering the unique node T = T(q) ∈ (0, 1) of the second Dirichlet eigenfunction as a nonlinear functional of potential q from the Lebesgue space L p [0, 1], we will study the optimization problems to minimize or to maximize T(q) subject to the constraint ||q|| L p = r. By applying the recent results on the differentiability and complete continuity of T(q) in q ∈ L p [0, 1], it will be proved that for the case 1 < p < ∞, these optimization problems are attained by some potentials. Moreover, a critical equation for optimizers will be derived. Finally, by considering the limit case p → 1, it will be found that the optimizers for the optimization problems for the case p = 1 are certain Dirac measures. These results are then used to deduce the optimal locations of nodes T(q).