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Optimization Related to Some Nonlocal Problems of Kirchhoff Type

Published online by Cambridge University Press:  20 November 2018

Behrouz Emamizadeh
Affiliation:
School of Mathematical Sciences, The University of Nottingham-Ningbo,199 Taikang East Road, Ningbo, 315100, China e-mail: Behrouz.Emamizadeh@nottingham.edu.cn
Amin Farjudian
Affiliation:
Center for Research on Embedded Systems, Halmstad University, Sweden e-mail: amin.farjudian@hh.se
Mohsen Zivari-Rezapour
Affiliation:
Department of Mathematics, Faculty of Mathematical and Computer Sciences, Shahid Chamran University, Golestan Blvd., Ahvaz, Iran e-mail: mzivari@scu.ac.ir
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Abstract

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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.

Type
Research Article
Copyright
Copyright © Canadian Mathematical Society 2016

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