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Minimizers for nonlocal perimeters of Minkowski type

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Abstract

We study a nonlocal perimeter functional inspired by the Minkowski content, whose main feature is that it interpolates between the classical perimeter and the volume functional. This nonlocal functionals arise in concrete applications, since the nonlocal character of the problems and the different behaviors of the energy at different scales allow the preservation of details and irregularities of the image in the process of removing white noises, thus improving the quality of the image without losing relevant features. In this paper, we provide a series of results concerning existence, rigidity and classification of minimizers, compactness results, isoperimetric inequalities, Poincaré–Wirtinger inequalities and density estimates. Furthermore, we provide the construction of planelike minimizers for this generalized perimeter under a small and periodic volume perturbation.

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Authors and Affiliations

  1. Dipartimento di Scienze Statistiche, Università di Padova, Via Battisti 241/243, 35121, Padua, Italy

    Annalisa Cesaroni

  2. Dipartimento di Matematica, Università di Milano, Via Saldini 50, 20133, Milan, Italy

    Serena Dipierro & Enrico Valdinoci

  3. Dipartimento di Matematica, Università di Pisa, Largo Pontecorvo 5, 56127, Pisa, Italy

    Matteo Novaga

  4. School of Mathematics and Statistics, University of Melbourne, 813 Swanston St, Parkville, VIC, 3010, Australia

    Enrico Valdinoci

  5. IMATI-CNR, Via Ferrata 1, 27100, Pavia, Italy

    Enrico Valdinoci

Authors
  1. Annalisa Cesaroni
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  2. Serena Dipierro
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  3. Matteo Novaga
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  4. Enrico Valdinoci
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Corresponding author

Correspondence to Matteo Novaga.

Additional information

Communicated by L. Ambrosio.

This work has been supported by the Andrew Sisson Fund 2017 and the Australian Research Council Discovery Project N.E.W. Nonlocal Equations at Work.

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Cesaroni, A., Dipierro, S., Novaga, M. et al. Minimizers for nonlocal perimeters of Minkowski type. Calc. Var. 57, 64 (2018). https://doi.org/10.1007/s00526-018-1335-9

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