Computing the First Eigenpair of the p-Laplacian via Inverse Iteration of Sublinear Supersolutions

Authors:
Rodney Josué Biezuner;Jed Brown;Grey Ercole;Eder Marinho Martins
Affiliations:
Departamento de Matemática--ICEx, Universidade Federal de Minas Gerais, Belo Horizonte, Brazil 30161-970;Laboratory of Hydraulics, Hydrology, and Glaciology (VAW), ETH Zürich, Zürich, Switzerland 8092 and Mathematics and Computer Science Division, Argonne National Laboratory, Argonne, USA 6 ...;Departamento de Matemática--ICEx, Universidade Federal de Minas Gerais, Belo Horizonte, Brazil 30161-970;Departamento de Matemática--ICEB, Universidade Federal de Ouro Preto, Ouro Preto, Brazil 35400-000
Venue:
Journal of Scientific Computing
Year:
2012

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Abstract

We introduce an iterative method for computing the first eigenpair (驴 p ,e p ) for the p-Laplacian operator with homogeneous Dirichlet data as the limit of (μ q, u q ) as q驴p 驴, where u q is the positive solution of the sublinear Lane-Emden equation $-\Delta_{p}u_{q}=\mu_{q}u_{q}^{q-1}$ with the same boundary data. The method is shown to work for any smooth, bounded domain. Solutions to the Lane-Emden problem are obtained through inverse iteration of a super-solution which is derived from the solution to the torsional creep problem. Convergence of u q to e p is in the C 1-norm and the rate of convergence of μ q to 驴 p is at least O(p驴q). Numerical evidence is presented.