  We define a family of functionals, called p-oscillation functionals, that can
be interpreted as discrete versions of the classical total variation functional
for p=1 and of the p-Dirichlet functionals for p>1. We introduce the notion of
minimizers and prove existence of solutions to the Dirichlet problem. Finally
we provide a description of Class A minimizers (i.e. minimizers under compact
perturbations) in dimension 1.
