2020/05/15 by Idriss Mazari, Mazari, Idriss · 1 citation
Computer Science · Mathematics · #35J15 #35Q93 #47A75 #49Q10 #49R05 #Advanced Mathematical Modeling in Engineering #Analysis of PDEs (math.AP) #FOS: Mathematics #Nonlinear Partial Differential Equations #Numerical methods in inverse problems #Optimization and Control (math.OC)
paper · pdf · doi:10.48550/arxiv.2005.07417
openalex publication_date 2020/05/15 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28
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 \mathcal Lv:=-Δ-V with Dirichlet boundary conditions with respect to the potential V, under L1 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^*||L1(Ω)2 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.