A dynamic $p$-Laplacian

TL;DR

This paper introduces a dynamic p-Laplacian extending the dynamic Laplacian concept, linking spectral properties to geometric coherent sets, and develops numerical methods to analyze these structures in dynamical systems.

Contribution

It generalizes the dynamic Laplacian to a dynamic p-Laplacian, establishes its spectral properties, and provides numerical schemes for analyzing coherent sets in dynamical systems.

Findings

The first eigenvalue converges to a dynamic Cheeger constant as p approaches 1.

Eigenfunctions' level sets identify boundaries of coherent sets.

Numerical examples demonstrate the method's effectiveness in dynamical systems.

Abstract

We generalise the dynamic Laplacian introduced in (Froyland, 2015) to a dynamic $p$-Laplacian, in analogy to the generalisation of the standard $2$-Laplacian to the standard $p$-Laplacian for $p>1$. Spectral properties of the dynamic Laplacian are connected to the geometric problem of finding "coherent" sets with persistently small boundaries under dynamical evolution, and we show that the dynamic $p$-Laplacian shares similar geometric connections. In particular, we prove that the first eigenvalue of the dynamic $p$-Laplacian with Dirichlet boundary conditions exists and converges to a dynamic version of the Cheeger constant introduced in (Froyland, 2015) as $p\rightarrow 1$. We develop a numerical scheme to estimate the leading eigenfunctions of the (nonlinear) dynamic $p$-Laplacian, and through a series of examples we investigate the behaviour of the level sets of these eigenfunctions. These level sets define the boundaries of sets in the domain of the dynamics that remain coherent under the dynamical evolution.