# American Institute of Mathematical Sciences

December  2019, 39(12): 6785-6799. doi: 10.3934/dcds.2019231

## 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

To Luis Caffarelli, on the occasion of his 70th birthday

Received  June 2018 Revised  December 2018 Published  June 2019

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

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 $1$.

Citation: Annalisa Cesaroni, Serena Dipierro, Matteo Novaga, Enrico Valdinoci. Minimizers of the $p$-oscillation functional. Discrete & Continuous Dynamical Systems - A, 2019, 39 (12) : 6785-6799. doi: 10.3934/dcds.2019231
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