Optimization related to some nonlocal problems of Kirchhoff type
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 a...
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University of Toronto Press
2016
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| Online Access: | https://eprints.nottingham.ac.uk/56206/ |
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| author | Emamizadeh, Behrouz Farjudian, Amin Zivari-Rezapour, Mohsen |
| author_facet | Emamizadeh, Behrouz Farjudian, Amin Zivari-Rezapour, Mohsen |
| author_sort | Emamizadeh, Behrouz |
| building | Nottingham Research Data Repository |
| collection | Online Access |
| description | 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. |
| first_indexed | 2025-11-14T20:33:21Z |
| format | Article |
| id | nottingham-56206 |
| institution | University of Nottingham Malaysia Campus |
| institution_category | Local University |
| language | English |
| last_indexed | 2025-11-14T20:33:21Z |
| publishDate | 2016 |
| publisher | University of Toronto Press |
| recordtype | eprints |
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| spelling | nottingham-562062019-03-04T10:25:51Z https://eprints.nottingham.ac.uk/56206/ Optimization related to some nonlocal problems of Kirchhoff type Emamizadeh, Behrouz Farjudian, Amin Zivari-Rezapour, Mohsen 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. University of Toronto Press 2016-06-01 Article PeerReviewed application/pdf en https://eprints.nottingham.ac.uk/56206/1/2016-Emamizadeh_Farjudian_ZivariRezapour-Optim_NonLocal.pdf Emamizadeh, Behrouz, 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 Kirchhoff equation; Rearrangements of functions; Maximization; Existence; Optimality condition https://www.cambridge.org/core/journals/canadian-journal-of-mathematics/article/optimization-related-to-some-nonlocal-problems-of-kirchhoff-type/610C4EC434BC2483DCF1A00857371969 doi:10.4153/CJM-2015-040-9 doi:10.4153/CJM-2015-040-9 |
| spellingShingle | Kirchhoff equation; Rearrangements of functions; Maximization; Existence; Optimality condition Emamizadeh, Behrouz Farjudian, Amin Zivari-Rezapour, Mohsen Optimization related to some nonlocal problems of Kirchhoff type |
| title | Optimization related to some nonlocal problems of Kirchhoff type |
| title_full | Optimization related to some nonlocal problems of Kirchhoff type |
| title_fullStr | Optimization related to some nonlocal problems of Kirchhoff type |
| title_full_unstemmed | Optimization related to some nonlocal problems of Kirchhoff type |
| title_short | Optimization related to some nonlocal problems of Kirchhoff type |
| title_sort | optimization related to some nonlocal problems of kirchhoff type |
| topic | Kirchhoff equation; Rearrangements of functions; Maximization; Existence; Optimality condition |
| url | https://eprints.nottingham.ac.uk/56206/ https://eprints.nottingham.ac.uk/56206/ https://eprints.nottingham.ac.uk/56206/ |