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 $Ω\subset\mathbb{R}^n$, we analyze a scheme that allows us to approximate the smallest value the ratio $\int_Ω|Dψ|^p dx/\int_Ω|ψ|^p dx$ can assume for functions $ψ$ that vanish on $\partial Ω$. The scheme in question also provides a natural way to approximate minimizing $ψ$. 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.