We analyse the functional
defined on Lipschitz functions with homogeneous Dirichlet boundary conditions. Our analysis
is performed directly on the functional without the need to approximate with smooth
-norms.
We prove that its ground states coincide with multiples of the distance
function to the boundary of the domain. Furthermore, we compute the
-subdifferential
of
and characterize the distance function as the unique nonnegative eigenfunction of the
subdifferential operator. We also study properties of general eigenfunctions,
in particular their nodal sets. Furthermore, we prove that the distance
function can be computed as the asymptotic profile of the gradient flow of
and construct analytic solutions of fast marching type. In addition, we give
a geometric characterization of the extreme points of the unit ball of
.
Finally, we transfer many of these results to a discrete version of the functional
defined on a finite weighted graph. Here, we analyze properties of distance functions
on graphs and their gradients. The main difference between the continuum and
discrete setting is that the distance function is not the unique nonnegative
eigenfunction on a graph.