Optimal Potentials of Measure Differential Equations with Given Spectral Data

Wen, Zhiyuan; Zhou, Lijuan; Zhang, Meirong

doi:10.1007/s10957-018-01462-y

Cited by 3 publications

(4 citation statements)

References 18 publications

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“…It is worth mentioning that, after the obtention of CC and CD of eigenvalues and eigenfunctions in potentials and in weights, the direct variational method has been applied to solve a series of optimization problems on eigenvalues of different types of differential equations. Besides the early papers of Wei et al 10 and Zhang 11 of ours, see also previous works 19–32 . In fact, such an approach to the optimization problems on eigenvalues is relatively new and is very fruitful.…”

Section: Summary and Further Problemsmentioning

confidence: 71%

“…Besides the early papers of Wei et al 10 and Zhang 11 of ours, see also previous works. [19][20][21][22][23][24][25][26][27][28][29][30][31][32] In fact, such an approach to the optimization problems on eigenvalues is relatively new and is very fruitful. Besides, the answers to these optimization problems also provide answers to their inverse problems, like (1.11).…”

Section: Summary and Further Problemsmentioning

confidence: 99%

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

Guo

Zhang

2022

Math Methods in App Sciences

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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).

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Section: Summary and Further Problemsmentioning

confidence: 71%

Section: Summary and Further Problemsmentioning

confidence: 99%

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

Guo

Zhang

2022

Math Methods in App Sciences

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“…Even for regular Sturm-Liouville problems (SLPs), the corresponding optimization problem of weights or potentials were also frequently related to Dirac delta functions (cf. other studies [10][11][12][13][14][15][16] ). More to the point, in physics, it is very important and common when the coefficients of second-order ordinary differential equations contain Dirac distributions.…”

Section: Introductionmentioning

confidence: 91%

The number of Dirac‐weighted eigenvalues of Sturm–Liouville equations with integrable potentials and an application to inverse problems

Chen

2021

Math Methods in App Sciences

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In this paper, we further Meirong Zhang et al.'s work by computing the number of weighted eigenvalues for Sturm–Liouville equations, equipped with general integrable potentials and Dirac weights, under Dirichlet boundary condition. We show that, for a Sturm–Liouville equation with a general integrable potential, if its weight is a positive linear combination of n Dirac delta functions, then it has at most n (may be less than n, or even be 0) distinct real Dirichlet eigenvalues, or every complex number is a Dirichlet eigenvalue; in particular, under some sharp condition, the number of Dirichlet eigenvalues is exactly n. Our main method is to introduce the concepts of characteristics matrix and characteristics polynomial for Sturm–Liouville problem with Dirac weights and put forward a general and direct algorithm used for computing eigenvalues. As an application, a class of inverse Dirichlet problems for Sturm–Liouville equations involving single Dirac distribution weights is studied.

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“…Even for regular Sturm-Liouville problems, the corresponding optimization problem of weights or potentials were also frequently related to Dirac Delta functions (cf. [8,9,10,13,18,21,23] etc). More to the point, in physics, it is very important and common when the coefficients of second-order ordinary differential equations contain Dirac distributions.…”

Section: Introductionmentioning

confidence: 99%

The number of Dirac-weighted eigenvalues of Sturm-Liouville equations with integrable potentials and an application to inverse problems

Chen,

2020

Preprint

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In this paper, we further Meirong Zhang, et al.'s work by computing the number of weighted eigenvalues for Sturm-Liouville equations, equipped with general integrable potentials and Dirac weights, under Dirichlet boundary condition. We show that, for a Sturm-Liouville equation with a general integrable potential, if its weight is a positive linear combination of n Dirac Delta functions, then it has at most n (may be less than n, or even be 0) distinct real Dirichlet eigenvalues, or every complex number is a Dirichlet eigenvalue; in particular, under some sharp condition, the number of Dirichlet eigenvalues is exactly n. Our main method is to introduce the concepts of characteristics matrix and characteristics polynomial for Sturm-Liouville problem with Dirac weights, and put forward a general and direct algorithm used for computing eigenvalues. As an application, a class of inverse Dirichelt problems for Sturm-Liouville equations involving single Dirac distribution weights is studied.

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Optimal Potentials of Measure Differential Equations with Given Spectral Data

Cited by 3 publications

References 18 publications

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

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

The number of Dirac‐weighted eigenvalues of Sturm–Liouville equations with integrable potentials and an application to inverse problems

The number of Dirac-weighted eigenvalues of Sturm-Liouville equations with integrable potentials and an application to inverse problems

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