Some results on radial symmetry in partial differential equations - Farjudian, Amin; Emamizadeh, Behrouz (NYJM, 2014)
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.