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

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 $q\rightarrow p^{-}$, where $u_{q}$ is the positive solution of the sublinear Lane-Emden equation $-Δ_{p}u_{q}=μ_{q}u_{q}^{q-1}$ 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 $μ_{q}$ to $λ_{p}$ is at least $O(p-q)$. Numerical evidence is presented.