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  • 1.
    Emamizadeh, Behrouz
    et al.
    School of Mathematical Sciences, The University of Nottingham Ningbo, Ningbo, China.
    Farjudian, Amin
    Halmstad University, School of Information Technology, Halmstad Embedded and Intelligent Systems Research (EIS), Centre for Research on Embedded Systems (CERES).
    Zivari-Rezapour, Mohsen
    Department of Mathematics, Faculty of Mathematical Sciences & Computer, Shahid Chamran University, Ahvaz, Iran.
    Optimization Related to Some Nonlocal Problems of Kirchhoff Type2016In: Canadian Journal of Mathematics - Journal Canadien de Mathematiques, ISSN 0008-414X, E-ISSN 1496-2479, Vol. 68, no 3, p. 521-540Article in journal (Refereed)
    Abstract [en]

    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, which is stable. Some numerical results are included to confirm the theory.

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