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Converse symmetry and Intermediate energy values in rearrangement optimization problems

Liu, Yichen; Emamizadeh, Behrouz


Yichen Liu

Behrouz Emamizadeh


This paper discusses three rearrangement optimization problems where the energy functional is connected with the Dirichlet or Robin boundary value problems. First, we consider a simple model of Dirichlet type, derive a symmetry result, and prove an intermediate energy theorem. For this model, we show that if the optimal domain (or its complement) is a ball centered at the origin, then the original domain must be a ball. As for the intermediate energy theorem, we show that if $\alpha,\beta$ denote the optimal values of corresponding minimization and maximization problems, respectively, then every $\gamma$ in $(\alpha,\beta)$ is achieved by solving a max-min problem. Second, we investigate a similar symmetry problem for the Dirichlet problems where the energy functional is nonlinear. Finally, we show the existence and uniqueness of rearrangement minimization problems associated with the Robin problems. In addition, we shall obtain a symmetry and a related asymptotic result.


Liu, Y., & Emamizadeh, B. (in press). Converse symmetry and Intermediate energy values in rearrangement optimization problems. SIAM Journal on Control and Optimization, 55(3),

Journal Article Type Article
Acceptance Date Apr 20, 2017
Online Publication Date Jun 27, 2017
Deposit Date Oct 12, 2017
Publicly Available Date Oct 12, 2017
Journal SIAM Journal on Control and Optimization
Print ISSN 0363-0129
Electronic ISSN 1095-7138
Publisher Society for Industrial and Applied Mathematics
Peer Reviewed Peer Reviewed
Volume 55
Issue 3
Keywords rearrangements, optimal solutions, symmetry, energy values, Robin problems, asymptotic
Public URL
Publisher URL


Converse Symmetry and Intermediate Energy Values in Rearrangement Optimization Problems. 2017.pdf (254 Kb)

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