Inverse iteration for $p$-ground states

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 $p\to \infty$ and thereby fashions a new approximation method for ground states of the infinity Laplacian.