Non-localization of eigenfunctions for Sturm-Liouville operators and applications

摘要

In this article, we investigate a non-localization property of the eigenfunctions of Sturm-Liouville operators A(a)=-partial derivative(xx) + a(.) Id with Dirichlet boundary conditions, where a(.) runs over the bounded nonnegative potential functions on the interval (0, L) with L 0. More precisely, we address the extremal spectral problem of minimizing the L-2-norm of a function e(.) on a measurable subset omega of (0, L), where e(.) runs over all eigenfunctions of A(a), at the same time with respect to all subsets omega having a prescribed measure and all L-infinity potential functions a(.) having a prescribed essentially upper bound. We provide some existence and qualitative properties of the minimizers, as well as precise lower and upper estimates on the optimal value. Several consequences in control and stabilization theory are then highlighted. (C) 2017 Elsevier Inc. All rights reserved.