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 $.
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