## Differential and Integral Equations

### On the Rayleigh quotient and the first eigenvalue for some vector-valued variational problems

#### Abstract

We prove that the first eigenvalue of a vector-valued $p$-Laplacian problem is equal to the first eigenvalue of the corresponding scalar $p$-Laplacian, and that the components of its first eigenvectors are merely copies of the first eigenfunction of the scalar problem. We also show variants of this result for some other homogeneous vector-valued problems.

#### Article information

Source
Differential Integral Equations, Volume 20, Number 4 (2007), 429-443.

Dates
First available in Project Euclid: 20 December 2012