Computing the first eigenpair of the p-Laplacian in annuli.
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Date
2015
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Abstract
We propose a method for computing the first eigenpair of the Dirichlet p-Laplacian,
p > 1, in the annulus Ωa,b = {x ∈ RN : a < |x| < b}, N > 1. For each t ∈ (a, b), we
use an inverse iteration method to solve two radial eigenvalue problems: one in the
annulus Ωa,t, with the corresponding eigenvalue λ−(t) and boundary conditions
u(a) = 0 = u
(t); and the other in the annulus Ωt,b, with the corresponding
eigenvalue λ+(t) and boundary conditions u
(t) = 0 = u(b). Next, we adjust
the parameter t using a matching procedure to make λ−(t) coincide with λ+(t),
thereby obtaining the first eigenvalue λp. Hence, by a simple splicing argument,
we obtain the positive, L∞-normalized, radial first eigenfunction up. The matching
parameter is the maximum point ρ of up. In order to apply this method, we derive
estimates for λ−(t) and λ+(t), and we prove that these functions are monotone
and (locally Lipschitz) continuous. Moreover, we derive upper and lower estimates
for the maximum point ρ, which we use in the matching procedure, and we also
present a direct proof that up converges to the L∞-normalized distance function to
the boundary as p → ∞. We also present some numerical results obtained using
this method.
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Keywords
Annulus, First eigenpair, Inverse iteration method, p-Laplacian
Citation
ERCOLE, G.; ESPÍRITO SANTO, J. C. do.; MARTINS, E. M. Computing the first eigenpair of the p-Laplacian in annuli. Journal of Mathematical Analysis and Applications, v. 422, p. 1277-1307, 2015. Disponível em: <http://www.sciencedirect.com/science/article/pii/S0022247X14008403>. Acesso em: 23 mar. 2017.