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