Some results on radial symmetry in partial differential equations

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

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Abstract

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.

Item Type: Article
Additional Information: Date of acceptance estimated
Keywords: Equality case; Faber-Krahn inequality; Principal eigenvalue; p-Laplace; Domain derivative; Pohozaev identity; Maximization; Volume constraint; Hamilton-Jacobi system
Schools/Departments: University of Nottingham Ningbo China > Faculty of Science and Engineering > School of Mathematical Sciences
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http://nyjm.albany.edu/j/2014/20-15.htmlPublisher
Depositing User: Yu, Tiffany
Date Deposited: 28 Feb 2019 09:10
Last Modified: 28 Feb 2019 09:10
URI: http://eprints.nottingham.ac.uk/id/eprint/56193

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