Research Repository

See what's under the surface

# Converse symmetry and Intermediate energy values in rearrangement optimization problems

## Authors

Yichen Liu yichen.liu07@yahoo.com

### Abstract

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.

Journal Article Type Article SIAM Journal on Control and Optimization 0363-0129 1095-7138 Society for Industrial and Applied Mathematics Peer Reviewed 55 3 Liu, Y., & Emamizadeh, B. (in press). Converse symmetry and Intermediate energy values in rearrangement optimization problems. SIAM Journal on Control and Optimization, 55(3), doi:10.1137/16M1100307 https://doi.org/10.1137/16M1100307 rearrangements, optimal solutions, symmetry, energy values, Robin problems, asymptotic http://epubs.siam.org/doi/10.1137/16M1100307 Copyright information regarding this work can be found at the following address: http://eprints.nottingh.../end_user_agreement.pdf

#### Files

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