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

TL;DR

This paper presents an iterative method to compute the first eigenpair of the p-Laplacian using inverse iteration of sublinear supersolutions, with proven convergence and numerical validation.

Contribution

The paper introduces a novel inverse iteration approach for the p-Laplacian eigenproblem, connecting solutions of Lane-Emden equations to eigenpairs.

Findings

Convergence of solutions to the Lane-Emden problem in the $C^{1}$-norm.

Rate of convergence of $$ to the eigenfunction is at least $O(p-q)$.

Numerical experiments support theoretical results.

Abstract

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.