Computing the first eigenpair of the p-Laplacian via inverse iteration of sublinear supersolutions

Details Computing the first eigenpair of the p-Laplacian via inverse iteration of sublinear supersolutions Computing the first eigenpair of the p-Laplacian via inverse iteration of sublinear supersolutions

2010-11-14

arxiv.org/abs/1011.3172v2

Mathematics

Analysis of PDEs

Section 5 was rewritten. Jed Brown was added as author

Scientific paper

10.1007/s10915-011-9540-0

We introduce an iterative method for computing the first eigenpair $(\lambda_{p},e_{p})$ for the $p$-Laplacian operator with homogeneous Dirichlet data as the limit of $(\mu_{q,}u_{q}) $ as $q\rightarrow p^{-}$, where $u_{q}$ is the positive solution of the sublinear Lane-Emden equation $-\Delta_{p}u_{q}=\mu_{q}u_{q}^{q-1}$ with 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 $\mu_{q}$ to $\lambda_{p}$ is at least $O(p-q)$. Numerical evidence is presented.

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