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Preface
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 |
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 $.
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. |
| [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. |
| [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. |
| [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. |
| [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. |
| [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. |
| [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. |
| [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. |
| [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. |
| [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. |
| [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. |
| [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. |
| [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. |
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. |
| [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. |
| [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. |
| [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. |
| [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. |
| [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. |
| [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. |
| [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. |
| [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. |
| [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. |
| [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. |
| [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. |
| [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. |
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