Some results on radial symmetry in partial differential equations
In this paper we will discuss three different problems which share the same conclusions. In the first one we revisit the well known Faber-Krahn inequality for the principal eigenvalue of the p-Laplace operator with zero homogeneous Dirichlet boundary conditions. Motivated by Chatelain, Choulli, and...
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| Format: | Article |
| Language: | English |
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NYJM
2014
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| Online Access: | https://eprints.nottingham.ac.uk/56193/ |
| _version_ | 1848799290560872448 |
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| author | Farjudian, Amin Emamizadeh, Behrouz |
| author_facet | Farjudian, Amin Emamizadeh, Behrouz |
| author_sort | Farjudian, Amin |
| building | Nottingham Research Data Repository |
| collection | Online Access |
| description | In this paper we will discuss three different problems which share the same conclusions. In the first one we revisit the well known Faber-Krahn inequality for the principal eigenvalue of the p-Laplace operator with zero homogeneous Dirichlet boundary conditions. Motivated by Chatelain, Choulli, and Henrot, 1996, we show in case the equality holds in the Faber-Krahn inequality, the domain of interest must be a ball. In the second problem we consider a generalization of the well known torsion problem and accordingly define a quantity that we name the p-torsional rigidity of the domain of interest. We maximize this quantity relative to a set of domains having the same volume, and prove that the optimal domain is a ball. The last problem is very similar in spirit to the second one. We consider a Hamilton-Jacobi boundary value problem, and define a quantity to be maximized relative to a set of domains having fixed volume. Again, we prove that the optimal domain is a ball. The main tools in our analysis are the method of domain derivatives, an appropriate generalized version of the Pohozaev identity, and the classical symmetrization techniques. |
| first_indexed | 2025-11-14T20:33:19Z |
| format | Article |
| id | nottingham-56193 |
| institution | University of Nottingham Malaysia Campus |
| institution_category | Local University |
| language | English |
| last_indexed | 2025-11-14T20:33:19Z |
| publishDate | 2014 |
| publisher | NYJM |
| recordtype | eprints |
| repository_type | Digital Repository |
| spelling | nottingham-561932019-02-28T09:10:32Z https://eprints.nottingham.ac.uk/56193/ Some results on radial symmetry in partial differential equations Farjudian, Amin Emamizadeh, Behrouz In this paper we will discuss three different problems which share the same conclusions. In the first one we revisit the well known Faber-Krahn inequality for the principal eigenvalue of the p-Laplace operator with zero homogeneous Dirichlet boundary conditions. Motivated by Chatelain, Choulli, and Henrot, 1996, we show in case the equality holds in the Faber-Krahn inequality, the domain of interest must be a ball. In the second problem we consider a generalization of the well known torsion problem and accordingly define a quantity that we name the p-torsional rigidity of the domain of interest. We maximize this quantity relative to a set of domains having the same volume, and prove that the optimal domain is a ball. The last problem is very similar in spirit to the second one. We consider a Hamilton-Jacobi boundary value problem, and define a quantity to be maximized relative to a set of domains having fixed volume. Again, we prove that the optimal domain is a ball. The main tools in our analysis are the method of domain derivatives, an appropriate generalized version of the Pohozaev identity, and the classical symmetrization techniques. NYJM 2014-03-17 Article PeerReviewed application/pdf en https://eprints.nottingham.ac.uk/56193/1/2014-Farjudian_Emamizadeh-Some_results_on_radial_symmetry_in_partial_differential_equations.pdf Farjudian, Amin and Emamizadeh, Behrouz (2014) Some results on radial symmetry in partial differential equations. New York Journal of Mathematics, 20 . pp. 241-255. ISSN 1076-9803 Equality case; Faber-Krahn inequality; Principal eigenvalue; p-Laplace; Domain derivative; Pohozaev identity; Maximization; Volume constraint; Hamilton-Jacobi system http://nyjm.albany.edu/j/2014/20-15.html |
| spellingShingle | Equality case; Faber-Krahn inequality; Principal eigenvalue; p-Laplace; Domain derivative; Pohozaev identity; Maximization; Volume constraint; Hamilton-Jacobi system Farjudian, Amin Emamizadeh, Behrouz Some results on radial symmetry in partial differential equations |
| title | Some results on radial symmetry in partial differential equations |
| title_full | Some results on radial symmetry in partial differential equations |
| title_fullStr | Some results on radial symmetry in partial differential equations |
| title_full_unstemmed | Some results on radial symmetry in partial differential equations |
| title_short | Some results on radial symmetry in partial differential equations |
| title_sort | some results on radial symmetry in partial differential equations |
| topic | Equality case; Faber-Krahn inequality; Principal eigenvalue; p-Laplace; Domain derivative; Pohozaev identity; Maximization; Volume constraint; Hamilton-Jacobi system |
| url | https://eprints.nottingham.ac.uk/56193/ https://eprints.nottingham.ac.uk/56193/ |