Article ID | Journal | Published Year | Pages | File Type |
---|---|---|---|---|
6418186 | Journal of Mathematical Analysis and Applications | 2015 | 31 Pages |
We propose a method for computing the first eigenpair of the Dirichlet p-Laplacian, p>1, in the annulus Ωa,b={xâRN:a<|x|1. For each tâ(a,b), we use an inverse iteration method to solve two radial eigenvalue problems: one in the annulus Ωa,t, with the corresponding eigenvalue λâ(t) and boundary conditions u(a)=0=uâ²(t); and the other in the annulus Ωt,b, with the corresponding eigenvalue λ+(t) and boundary conditions uâ²(t)=0=u(b). Next, we adjust the parameter t using a matching procedure to make λâ(t) coincide with λ+(t), thereby obtaining the first eigenvalue λp. Hence, by a simple splicing argument, we obtain the positive, Lâ-normalized, radial first eigenfunction up. The matching parameter is the maximum point Ï of up. In order to apply this method, we derive estimates for λâ(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 Ï, which we use in the matching procedure, and we also present a direct proof that up converges to the Lâ-normalized distance function to the boundary as pââ. We also present some numerical results obtained using this method.