Improved energy bounds for Schrödinger operators

Brasco, Lorenzo; Buttazzo, Giuseppe

doi:10.1007/s00526-014-0774-1

Cited by 4 publications

(6 citation statements)

References 25 publications

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“…Such quantitative inequalities have been studied in great details using shape optimisation when the optimisation parameter is the set Ω itself rather than the potential V [9,26]. Let us briefly mention that this inequality was established in the case Ω = B(0; 1) in [34], and we also mention [8,12], where quantitative inequalities for potential optimisation problems are obtained in other settings, for L p constraints. In this article, due to the particular nature of our constraints, our methods are significantly different from [8,12].…”

Section: Statement Of the Resultsmentioning

confidence: 99%

Quantitative Stability for Eigenvalues of Schrödinger Operator, Quantitative Bathtub Principle, and Application to the Turnpike Property for a Bilinear Optimal Control Problem

Mazari¹,

Ruiz-Balet²

2022

SIAM J. Math. Anal.

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This work is concerned with two optimisation problems that we tackle from a qualitative perspective. The first one deals with quantitative inequalities for spectral optimisation problems for Schrödinger operators in general domains, the second one deals with the turnpike property for optimal bilinear control problems. In the first part of this article, we prove, under mild technical assumptions, quantitative inequalities for the optimisation of the first eigenvalue of −∆−V with Dirichlet boundary conditions with respect to the potential V , under L ∞ and L 1 constraints. This is done using a new method of proof which relies on in a crucial way on a quantitative bathtub principle. We believe our approach susceptible of being generalised to other steady elliptic optimisation problems. In the second part of this paper, we use this inequality to tackle a turnpike problem. Namely, considering a bilinear control system of the form ut − ∆u = Vu, V = V(t, x) being the control, can we give qualitative information, under L ∞ and L 1 constraints on V, on the solutions of the optimisation problem sup ´Ω u(T, x)dx? We prove that the quantitative inequality for eigenvalues implies an integral turnpike property: defining I * as the set of optimal potentials for the eigenvalue optimisation problem and V * T as a solution of the bilinear optimal control problem, the quantity ´T 0 dist L 1 (V * T (t, •) , I * ) 2 is bounded uniformly in T .

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Section: Statement Of the Resultsmentioning

confidence: 99%

Quantitative Stability for Eigenvalues of Schrödinger Operator, Quantitative Bathtub Principle, and Application to the Turnpike Property for a Bilinear Optimal Control Problem

Mazari¹,

Ruiz-Balet²

2022

SIAM J. Math. Anal.

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“…Quantitative estimates for optimal potentials In the parametric context, that is, when optimising a criterion with respect to a potential, two references whose results are related to the one of the present paper are [10] and [16]; in both these papers, the L ∞ constraint 0 V 1 we consider in the present paper is not considered and they mainly deal with L p constraints. Namely, in [16], the main result, in the two dimensional case, is the following: consider, for a parameter γ > 0, a non-postivie potential V ∈ L 1+γ (IR 2 ) , V 0, the operator…”

Section: Quantitative Spectral Inequalitiesmentioning

confidence: 96%

“…In [10], a stability estimate for the Dirichlet energy with respect to the potential is obtained. One of their main theorems reads as follows [10, Theorem B]: let Ω be a smooth domain in IR n and f ∈ W −1,2 (Ω).…”

Section: Quantitative Spectral Inequalitiesmentioning

confidence: 99%

“…Here, the proof of this stability results relies on stability estimates for functional inequalities, and the optimality conditions are quite different from ours, as the optimality system in their case is a non-autonomous, semilinear equation. It is not clear to us that the constraints 0 V 1 we consider here, as well as the spectral quantity we are optimising, can be handled through the methods of [10].…”

Section: Quantitative Spectral Inequalitiesmentioning

confidence: 99%

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

Mazari

2020

Preprint

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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: iffor 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.

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“…To mention a few works, we point to the seminal [21] for the quantitative isoperimetric inequality, and to [9] for quantitative spectral inequalities. Regarding quantitative inequalities for (stationary) control problems we refer to [8] for a quantitative inequality for the natural Dirichlet energy, to [12] for a quantitative spectral inequality (with respect to the potential) in IR n (both these works are done under L p constraints), to [29] for a quantitative spectral inequality in the ball under L 1 and L ∞ constraints and to [31] for a generalisation of this inequality to other domains, and for an application to the turnpike property.…”

Section: Quantitative Inequalitiesmentioning

confidence: 99%

Quantitative estimates for parabolic optimal control problems under L∞ and L1 constraints

Mazari

2022

Nonlinear Analysis

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In this article, we present two different approaches for obtaining quantitative inequalities in the context of parabolic optimal control problems. Our model consists of a linearly controlled heat equation with Dirichlet boundary condition (u f )t − ∆u f = f , f being the control. We seek to maximise the functional JT (f ) := 1 2 ˜(0;T )×Ω u 2 f or, for some ε > 0 ,and to obtain quantitative estimates for these maximisation problems. We offer two approaches in the case where the domain Ω is a ball. In that case, if f satisfies L 1 and L ∞ constraints and does not depend on time, we propose a shape derivative approach that shows that, for any competitor f = f (x) satisfying the same constraints, we have JT, f * being the maximiser. Through our proof of this time-independent case, we also show how to obtain coercivity norms for shape hessians in such parabolic optimisation problems. We also consider the case where f = f (t, x) satisfies a global L ∞ constraint and, for every t ∈ (0; T ), an L 1 constraint. In this case, assuming ε > 0, we prove an estimate of the formwhere aε(t) > 0 for any t ∈ (0; T ). The proof of this result relies on a uniform bathtub principle.

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Improved energy bounds for Schrödinger operators

Cited by 4 publications

References 25 publications

Quantitative Stability for Eigenvalues of Schrödinger Operator, Quantitative Bathtub Principle, and Application to the Turnpike Property for a Bilinear Optimal Control Problem

Quantitative Stability for Eigenvalues of Schrödinger Operator, Quantitative Bathtub Principle, and Application to the Turnpike Property for a Bilinear Optimal Control Problem

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

Quantitative estimates for parabolic optimal control problems under L∞ and L1 constraints

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