 

Amin Farjudian and Behrouz Emamizadeh
Some results on radial symmetry in partial differential equations view print


Published: 
March 17, 2014

Keywords: 
Equality case, FaberKrahn inequality, Principal eigenvalue, pLaplace, Domain derivative, Pohozaev identity, Maximization, Volume constraint, HamiltonJacobi system 
Subject: 
35J62, 35P30, 35F21 


Abstract
In this paper we will discuss three different problems which share
the same conclusions. In the first one we revisit the well known
FaberKrahn inequality for the principal eigenvalue of the
pLaplace operator with zero homogeneous Dirichlet boundary
conditions. Motivated by Chatelain, Choulli, and
Henrot, 1996, we
show in case the equality holds in the FaberKrahn 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 ptorsional 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 HamiltonJacobi 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.


Author information
Faculty of Science and Engineering, University of Nottingham Ningbo China, 199 Taikang East Road, Ningbo, 315100, China
Amin.Farjudian@nottingham.edu.cn
Behrouz.Emamizadeh@nottingham.edu.cn

