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

Minimizers of the $ p $-oscillation functional

Annalisa Cesaroni 1, , Serena Dipierro 2,, , Matteo Novaga 3, and  Enrico Valdinoci 4,

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 $.

Keywords: $ p $-oscillation functional, Minkowski content, discrete total variation functional, one-dimensional solutions, minimizers.
Mathematics Subject Classification: Primary: 35B05, 49Q20; Secondary: 35B53.
Citation: Annalisa Cesaroni, Serena Dipierro, Matteo Novaga, Enrico Valdinoci. Minimizers of the $ p $-oscillation functional. Discrete & Continuous Dynamical Systems, 2019, 39 (12) : 6785-6799. doi: 10.3934/dcds.2019231
References:
[1]

L. Ambrosio, N. Fusco and D. Pallara, Functions of Bounded Variation and Free Discontinuity Problems, Oxford Mathematical Monographs, The Clarendon Press, Oxford University Press, New York, 2000.  Google Scholar

[2]

M. Barchiesi, S. H. Kang, T. M. Le, M. Morini and M. Ponsiglione, A variational model for infinite perimeter segmentations based on Lipschitz level set functions: Denoising while keeping finely oscillatory boundaries, Multiscale Model. Simul., 8 (2010), 1715-1741.  doi: 10.1137/090773659.  Google Scholar

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H. H. Bauschke and P. L. Combettes, Convex analysis and monotone operator theory in Hilbert spaces, CMS Books in Mathematics/Ouvrages de Mathématiques de la SMC, With a foreword by Hédy Attouch, Springer, New York, 2011. doi: 10.1007/978-1-4419-9467-7.  Google Scholar

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A. Cesaroni, S. Dipierro, M. Novaga and E. Valdinoci, Minimizers for nonlocal perimeters of Minkowski type, Calc. Var. Partial Differential Equations, 57, (2018), Art. 64, 40. doi: 10.1007/s00526-018-1335-9.  Google Scholar

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A. Cesaroni and M. Novaga, Isoperimetric problems for a nonlocal perimeter of Minkowski type, Geom. Flows, 2 (2017), 86-93.  doi: 10.1515/geofl-2017-0003.  Google Scholar

[6]

A. Chambolle, A. Giacomini and L. Lussardi, Continuous limits of discrete perimeters, M2AN Math. Model. Numer. Anal., 44, (2010), 207–230. doi: 10.1051/m2an/2009044.  Google Scholar

[7]

A. Chambolle, S. Lisini and L. Lussardi, A remark on the anisotropic outer Minkowski content, Adv. Calc. Var., 7 (2014), 241-266.  doi: 10.1515/acv-2013-0103.  Google Scholar

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A. Chambolle, M. Morini and M. Ponsiglione, A nonlocal mean curvature flow and its semi-implicit time-discrete approximation, SIAM J. Math. Anal., 44 (2012), 4048-4077.  doi: 10.1137/120863587.  Google Scholar

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A. Chambolle, M. Morini and M. Ponsiglione, Nonlocal curvature flows, Arch. Ration. Mech. Anal., 218 (2015), 1263-1329.  doi: 10.1007/s00205-015-0880-z.  Google Scholar

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R. Courant, Dirichlet's Principle, Conformal Mapping, and Minimal Surfaces, With an appendix by M. Schiffer; Reprint of the 1950 original, Springer-Verlag, New York-Heidelberg, 1977.  Google Scholar

[11]

S. Dipierro, M. Novaga and E. Valdinoci, On a Minkowski geometric flow in the plane: Evolution of curves with lack of scale invariance, J. Lond. Math. Soc. (2), 99 (2019), 31-51.  doi: 10.1112/jlms.12162.  Google Scholar

[12]

M. Novaga and E. Paolini, Regularity results for some 1-homogeneous functionals, Nonlinear Anal. Real World Appl., 3 (2002), 555-566.  doi: 10.1016/S1468-1218(01)00048-7.  Google Scholar

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E. Valdinoci, A fractional framework for perimeters and phase transitions, Milan J. Math., 81 (2013), 1-23.  doi: 10.1007/s00032-013-0199-x.  Google Scholar

show all references

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

References:
[1]

L. Ambrosio, N. Fusco and D. Pallara, Functions of Bounded Variation and Free Discontinuity Problems, Oxford Mathematical Monographs, The Clarendon Press, Oxford University Press, New York, 2000.  Google Scholar

[2]

M. Barchiesi, S. H. Kang, T. M. Le, M. Morini and M. Ponsiglione, A variational model for infinite perimeter segmentations based on Lipschitz level set functions: Denoising while keeping finely oscillatory boundaries, Multiscale Model. Simul., 8 (2010), 1715-1741.  doi: 10.1137/090773659.  Google Scholar

[3]

H. H. Bauschke and P. L. Combettes, Convex analysis and monotone operator theory in Hilbert spaces, CMS Books in Mathematics/Ouvrages de Mathématiques de la SMC, With a foreword by Hédy Attouch, Springer, New York, 2011. doi: 10.1007/978-1-4419-9467-7.  Google Scholar

[4]

A. Cesaroni, S. Dipierro, M. Novaga and E. Valdinoci, Minimizers for nonlocal perimeters of Minkowski type, Calc. Var. Partial Differential Equations, 57, (2018), Art. 64, 40. doi: 10.1007/s00526-018-1335-9.  Google Scholar

[5]

A. Cesaroni and M. Novaga, Isoperimetric problems for a nonlocal perimeter of Minkowski type, Geom. Flows, 2 (2017), 86-93.  doi: 10.1515/geofl-2017-0003.  Google Scholar

[6]

A. Chambolle, A. Giacomini and L. Lussardi, Continuous limits of discrete perimeters, M2AN Math. Model. Numer. Anal., 44, (2010), 207–230. doi: 10.1051/m2an/2009044.  Google Scholar

[7]

A. Chambolle, S. Lisini and L. Lussardi, A remark on the anisotropic outer Minkowski content, Adv. Calc. Var., 7 (2014), 241-266.  doi: 10.1515/acv-2013-0103.  Google Scholar

[8]

A. Chambolle, M. Morini and M. Ponsiglione, A nonlocal mean curvature flow and its semi-implicit time-discrete approximation, SIAM J. Math. Anal., 44 (2012), 4048-4077.  doi: 10.1137/120863587.  Google Scholar

[9]

A. Chambolle, M. Morini and M. Ponsiglione, Nonlocal curvature flows, Arch. Ration. Mech. Anal., 218 (2015), 1263-1329.  doi: 10.1007/s00205-015-0880-z.  Google Scholar

[10]

R. Courant, Dirichlet's Principle, Conformal Mapping, and Minimal Surfaces, With an appendix by M. Schiffer; Reprint of the 1950 original, Springer-Verlag, New York-Heidelberg, 1977.  Google Scholar

[11]

S. Dipierro, M. Novaga and E. Valdinoci, On a Minkowski geometric flow in the plane: Evolution of curves with lack of scale invariance, J. Lond. Math. Soc. (2), 99 (2019), 31-51.  doi: 10.1112/jlms.12162.  Google Scholar

[12]

M. Novaga and E. Paolini, Regularity results for some 1-homogeneous functionals, Nonlinear Anal. Real World Appl., 3 (2002), 555-566.  doi: 10.1016/S1468-1218(01)00048-7.  Google Scholar

[13]

E. Valdinoci, A fractional framework for perimeters and phase transitions, Milan J. Math., 81 (2013), 1-23.  doi: 10.1007/s00032-013-0199-x.  Google Scholar

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