Optimization related to some nonlocal problems of Kirchhoff type

Behrouz Emamizadeh, Amin Farjudian, Mohsen Zivari-Rezapour

Research output: Journal PublicationArticlepeer-review

2 Citations (Scopus)
3 Downloads (Pure)

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 confirm the theory.

Original languageEnglish
Pages (from-to)521-540
Number of pages20
JournalCanadian Journal of Mathematics
Volume68
Issue number3
DOIs
Publication statusPublished - Jun 2016

Keywords

  • Existence
  • Kirchhoff equation
  • Maximization
  • Optimality condition
  • Rearrangements of functions

ASJC Scopus subject areas

  • Mathematics (all)

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