Inverse iteration for $p$-ground states

TL;DR

This paper adapts inverse iteration methods to nonlinear PDE eigenvalue problems, providing new schemes for approximating ground states for various p-Laplacian problems and extending to the infinity Laplacian.

Contribution

It introduces a novel inverse iteration scheme for nonlinear PDE eigenvalue problems, including the p-Laplacian and infinity Laplacian, with convergence analysis.

Findings

Scheme effectively approximates the smallest eigenvalue ratio for p-Laplacian.

Method extends to the limit as p approaches infinity, approximating infinity Laplacian ground states.

Provides a natural way to approximate minimizing functions for the eigenvalue problem.

Abstract

We adapt the inverse iteration method for symmetric matrices to some nonlinear PDE eigenvalue problems. In particular, for $p\in (1,\infty)$ and a given domain $\Omega\subset\mathbb{R}^n$, we analyze a scheme that allows us to approximate the smallest value the ratio $\int_\Omega|D\psi|^p dx/\int_\Omega|\psi|^p dx$ can assume for functions $\psi$ that vanish on $\partial \Omega$. The scheme in question also provides a natural way to approximate minimizing $\psi$. Our analysis also extends in the limit as $p\rightarrow\infty$ and thereby fashions a new approximation method for ground states of the infinity Laplacian.