Quantitative inequality for the eigenvalue of a Schrödinger operator in the ball

Abstract : The aim of this article is to prove a quantitative inequality for the first eigenvalue of a Schrödinger operator in the ball. More precisely, we optimize the first eigenvalue λ(V) of the operator Lv := −∆ + V with Dirichlet boundary conditions with respect to the potential V , under L 1 and L ∞ constraints on V. The solution has been known to be the characteristic function of a centered ball, but this article aims at proving a sharp growth rate of the following form: if V * is a minimizer, then λ(V) − λ(V *) C||V − V * || 2 L 1 (Ω) for some C > 0. The proof relies on two notions of derivatives for shape optimization: parametric derivatives and shape derivatives. We use parametric derivatives to handle radial competitors, and shape derivatives to deal with normal deformation of the ball. A dichotomy is then established to extend the result to all other potentials. We develop a new method to handle radial distributions and a comparison principle to handle second order shape derivatives at the ball. Finally, we add some remarks regarding the coercivity norm of the second order shape derivative in this context.
Complete list of metadatas

Cited literature [43 references]  Display  Hide  Download

https://hal.archives-ouvertes.fr/hal-02280179
Contributor : Idriss Mazari <>
Submitted on : Wednesday, October 2, 2019 - 2:17:28 PM
Last modification on : Saturday, October 5, 2019 - 1:22:07 AM

File

Quantitative2019V2.pdf
Files produced by the author(s)

Identifiers

  • HAL Id : hal-02280179, version 2

Citation

Idriss Mazari. Quantitative inequality for the eigenvalue of a Schrödinger operator in the ball. 2019. ⟨hal-02280179v2⟩

Share

Metrics

Record views

89

Files downloads

52