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 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.