  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.
