On the Rayleigh quotient and the first eigenvalue for some vector-valued variational problems. (English) Zbl 1212.35098

Brock, Friedemann; Manásevich, Raul

Differ. Integral Equ. 20, No. 4, 429-444 (2007).

Summary: We prove that the first eigenvalue of a vector-valued \(p\)-Laplacian problem is equal to the first eigenvalue of the corresponding \(p\)-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.

MSC:

35J50	Variational methods for elliptic systems
35J70	Degenerate elliptic equations
49R05	Variational methods for eigenvalues of operators
47A75	Eigenvalue problems for linear operators

Keywords:

\(p\)-Laplacian; Rayleigh quotient; vector-valued function