## 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.

Original language | English |
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Pages (from-to) | 241-255 |

Number of pages | 15 |

Journal | New York Journal of Mathematics |

Volume | 20 |

Publication status | Published - 2014 |

Externally published | Yes |

## Keywords

- Domain derivative
- Equality case
- Faber-krahn inequality
- Hamilton-jacobi system
- Maximization
- P-laplace
- Pohozaev identity
- Principal eigenvalue
- Volume constraint

## ASJC Scopus subject areas

- General Mathematics