Minimizers of the p-oscillation functional

TL;DR

This paper introduces p-oscillation functionals as discrete analogs of total variation and p-Dirichlet functionals, establishing existence of minimizers and characterizing certain minimizers in one dimension.

Contribution

It defines p-oscillation functionals, proves the existence of minimizers for the Dirichlet problem, and characterizes Class A minimizers in one dimension.

Findings

Existence of solutions to the Dirichlet problem for p-oscillation functionals.

Characterization of Class A minimizers in dimension 1.

p-oscillation functionals generalize total variation and p-Dirichlet functionals.

Abstract

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.