Optimization related to some nonlocal problems of Kirchhoff type

Emamizadeh, Behrouz and Farjudian, Amin and Zivari-Rezapour, Mohsen (2016) Optimization related to some nonlocal problems of Kirchhoff type. Canadian Journal of Mathematics, 68 (03). pp. 521-540. ISSN 1496-4279

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Abstract

In this paper we introduce two rearrangement optimization problems, one being a maximization and the other a minimization problem, related to a nonlocal boundary value problem of Kirchhoff type. Using the theory of rearrangements as developed by G. R. Burton, we are able to show that both problems are solvable and derive the corresponding optimality conditions. These conditions in turn provide information concerning the locations of the optimal solutions.The strict convexity of the energy functional plays a crucial role in both problems. The popular case in which the rearrangement class (i.e., the admissible set) is generated by a characteristic function is also considered. We show that in this case, the maximization problem gives rise to a free boundary problem of obstacle type, which turns out to be unstable. On the other hand, the minimization problem leads to another free boundary problem of obstacle type that is stable. Some numerical results are included to conûrm the theory.

Item Type: Article
Additional Information: First published in Canadian Journal of Mathematics at https://doi.org/10.4153/CJM-2015-040-9. Copyright © 2016, Canadian Mathematical Society.
Keywords: Kirchhoff equation; Rearrangements of functions; Maximization; Existence; Optimality condition
Schools/Departments: University of Nottingham Ningbo China > Faculty of Science and Engineering > School of Mathematical Sciences
Identification Number: https://doi.org/10.4153/CJM-2015-040-9
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Depositing User: Yu, Tiffany
Date Deposited: 04 Mar 2019 10:25
Last Modified: 04 Mar 2019 10:25
URI: http://eprints.nottingham.ac.uk/id/eprint/56206

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