Differential and Integral Equations

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

Friedemann Brock and R. Manásevich

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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

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

First available in Project Euclid: 20 December 2012

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Mathematical Reviews number (MathSciNet)

Zentralblatt MATH identifier

Primary: 35J50: Variational methods for elliptic systems
Secondary: 35J55 35J60: Nonlinear elliptic equations 35P15: Estimation of eigenvalues, upper and lower bounds 35P30: Nonlinear eigenvalue problems, nonlinear spectral theory 49R50


Brock, Friedemann; Manásevich, R. On the Rayleigh quotient and the first eigenvalue for some vector-valued variational problems. Differential Integral Equations 20 (2007), no. 4, 429--443. https://projecteuclid.org/euclid.die/1356039461

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