CMS/SMC
Canadian Mathematical Society
www.cms.math.ca
Canadian Mathematical Society
  location:  PublicationsjournalsCJM
Abstract view

Optimization Related to Some Nonlocal Problems of Kirchhoff Type

  Published:2016-02-17
 Printed: Jun 2016
  • Behrouz Emamizadeh,
    School of Mathematical Sciences , The University of Nottingham-Ningbo , 199 Taikang East Road, Ningbo, 315100, China
  • Amin Farjudian,
    Center for Research on Embedded Systems , Halmstad University, Sweden
  • Mohsen Zivari-Rezapour,
    Department of Mathematics , Faculty of Mathematical and Computer Sciences , Shahid Chamran University , Golestan Blvd., Ahvaz, Iran
Format:   LaTeX   MathJax   PDF  

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, which is stable. Some numerical results are included to confirm the theory.
Keywords: Kirchhoff equation, rearrangements of functions, maximization, existence, optimality condition Kirchhoff equation, rearrangements of functions, maximization, existence, optimality condition
MSC Classifications: 35J20, 35J25 show english descriptions Variational methods for second-order elliptic equations
Boundary value problems for second-order elliptic equations
35J20 - Variational methods for second-order elliptic equations
35J25 - Boundary value problems for second-order elliptic equations
 

© Canadian Mathematical Society, 2017 : https://cms.math.ca/