Converse symmetry and Intermediate energy values in rearrangement optimization problems
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 sho...
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| Format: | Article |
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Society for Industrial and Applied Mathematics
2017
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| Online Access: | https://eprints.nottingham.ac.uk/47205/ |
| _version_ | 1848797489371545600 |
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| author | Liu, Yichen Emamizadeh, Behrouz |
| author_facet | Liu, Yichen Emamizadeh, Behrouz |
| author_sort | Liu, Yichen |
| building | Nottingham Research Data Repository |
| collection | Online Access |
| description | 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. |
| first_indexed | 2025-11-14T20:04:41Z |
| format | Article |
| id | nottingham-47205 |
| institution | University of Nottingham Malaysia Campus |
| institution_category | Local University |
| last_indexed | 2025-11-14T20:04:41Z |
| publishDate | 2017 |
| publisher | Society for Industrial and Applied Mathematics |
| recordtype | eprints |
| repository_type | Digital Repository |
| spelling | nottingham-472052020-05-04T18:51:40Z https://eprints.nottingham.ac.uk/47205/ Converse symmetry and Intermediate energy values in rearrangement optimization problems Liu, Yichen Emamizadeh, Behrouz 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. Society for Industrial and Applied Mathematics 2017-06-27 Article PeerReviewed Liu, Yichen and Emamizadeh, Behrouz (2017) Converse symmetry and Intermediate energy values in rearrangement optimization problems. SIAM Journal on Control and Optimization, 55 (3). pp. 2088-2107. ISSN 1095-7138 rearrangements optimal solutions symmetry energy values Robin problems asymptotic http://epubs.siam.org/doi/10.1137/16M1100307 doi:10.1137/16M1100307 doi:10.1137/16M1100307 |
| spellingShingle | rearrangements optimal solutions symmetry energy values Robin problems asymptotic Liu, Yichen Emamizadeh, Behrouz Converse symmetry and Intermediate energy values in rearrangement optimization problems |
| title | Converse symmetry and Intermediate energy values in rearrangement optimization problems |
| title_full | Converse symmetry and Intermediate energy values in rearrangement optimization problems |
| title_fullStr | Converse symmetry and Intermediate energy values in rearrangement optimization problems |
| title_full_unstemmed | Converse symmetry and Intermediate energy values in rearrangement optimization problems |
| title_short | Converse symmetry and Intermediate energy values in rearrangement optimization problems |
| title_sort | converse symmetry and intermediate energy values in rearrangement optimization problems |
| topic | rearrangements optimal solutions symmetry energy values Robin problems asymptotic |
| url | https://eprints.nottingham.ac.uk/47205/ https://eprints.nottingham.ac.uk/47205/ https://eprints.nottingham.ac.uk/47205/ |