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

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Abstract

We introduce an iterative method for computing the first eigenpair (λ p ,e p ) for the p-Laplacian operator with homogeneous Dirichlet data as the limit of (μ q, u q ) as q→p −, 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 λ p is at least O(p−q). Numerical evidence is presented.

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Authors and Affiliations

  1. Departamento de Matemática—ICEx, Universidade Federal de Minas Gerais, Av. Antônio Carlos 6627, Caixa Postal 702, 30161-970, Belo Horizonte, MG, Brazil

    Rodney Josué Biezuner & Grey Ercole

  2. Laboratory of Hydraulics, Hydrology, and Glaciology (VAW), ETH Zürich, 8092, Zürich, Switzerland

    Jed Brown

  3. Mathematics and Computer Science Division, Argonne National Laboratory, Argonne, IL, 60439, USA

    Jed Brown

  4. Departamento de Matemática—ICEB, Universidade Federal de Ouro Preto, Campus Universitário Morro do Cruzeiro, 35400-000, Ouro Preto, MG, Brazil

    Eder Marinho Martins

Authors
  1. Rodney Josué Biezuner
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  2. Jed Brown
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  3. Grey Ercole
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  4. Eder Marinho Martins
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Correspondence to Grey Ercole.

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Biezuner, R.J., Brown, J., Ercole, G. et al. Computing the First Eigenpair of the p-Laplacian via Inverse Iteration of Sublinear Supersolutions. J Sci Comput 52, 180–201 (2012). https://doi.org/10.1007/s10915-011-9540-0

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