Minimizers of the $ p $-oscillation functional

Annalisa Cesaroni; Serena Dipierro; Matteo Novaga; Enrico Valdinoci

doi:10.3934/dcds.2019231

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2019, Volume 39, Issue 12: 6785-6799. Doi: 10.3934/dcds.2019231

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Minimizers of the $ p $-oscillation functional

1.
Dipartimento di Scienze Statistiche, Università di Padova, Via Battisti 241/243, 35121 Padova, Italy
2.
Department of Mathematics and Statistics, University of Western Australia, 35 Stirling Hwy, Crawley WA 6009, Australia
3.
Dipartimento di Matematica, Università di Pisa, Largo Pontecorvo 5, 56127 Pisa, Italy
4.
Department of Mathematics and Statistics, University of Western Australia, 35 Stirling Hwy, Crawley WA 6009, Australia

^* Corresponding author: Serena Dipierro
^* Corresponding author: Serena Dipierro

Received: June 2018

Revised: December 2018

Published: September 2019

This work has been supported by the Australian Research Council grant "N.E.W." Nonlocal Equation at Work

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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 $.

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Mathematics Subject Classification: Primary: 35B05, 49Q20; Secondary: 35B53.

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