December  2015, 35(12): 5711-5723. doi: 10.3934/dcds.2015.35.5711

Eventual regularity for the parabolic minimal surface equation

1. 

Dipartimento di Matematica, Università di Roma Tor Vergata, Via della Ricerca Scientifica 1, 00133 Roma, Italy

2. 

Dipartimento di Matematica, Universitá di Pisa, Largo Bruno Pontecorvo 5, I-56127 Pisa

3. 

Dipartimento di Informatica, Università di Verona, Strada le Grazie 15, 37134 Verona, Italy

Received  January 2014 Published  May 2015

We show that the parabolic minimal surface equation has an eventual regularization effect, that is, the solution becomes smooth after a strictly positive finite time.
Citation: Giovanni Bellettini, Matteo Novaga, Giandomenico Orlandi. Eventual regularity for the parabolic minimal surface equation. Discrete and Continuous Dynamical Systems, 2015, 35 (12) : 5711-5723. doi: 10.3934/dcds.2015.35.5711
References:
[1]

L. Ambrosio, Corso Introduttivo alla Teoria Geometrica della Misura ed alle Superfici Minime, Edizioni della Scuola Normale, Pisa, 1997.

[2]

R. A. Adams, Sobolev Spaces, Academic Press, New York 1975.

[3]

F. Andreu, V. Caselles, J. I. Díaz and J. M. Mazón, Some qualitative properties for the total variation flow, J. Funct. Anal., 188 (2002), 516-547. doi: 10.1006/jfan.2001.3829.

[4]

F. Andreu, V. Caselles and J. M. Mazón, Parabolic Quasilinear Equations Minimizing Linear Growth Functionals, Oxford Mathematical Monographs, Birkhäuser, Basel, 2004. doi: 10.1007/978-3-0348-7928-6.

[5]

G. Anzellotti, Pairings between measures and bounded functions and compensated compactness, Ann. Mat. Pura Appl., 135 (1983), 293-318. doi: 10.1007/BF01781073.

[6]

G. Bellettini, Lecture Notes on Mean Curvature Flow, Barriers and Singular Perturbations, Edizioni della Scuola Normale, Pisa, 2013. doi: 10.1007/978-88-7642-429-8.

[7]

G. Bellettini, V. Caselles and M. Novaga, The total variation flow in $\mathbb{R}^N2$, J. Differential Equations, 184 (2002), 475-525. doi: 10.1006/jdeq.2001.4150.

[8]

G. Bellettini, V. Caselles and M. Novaga, Explicit solutions of the eigenvalue problem -div$(\frac{Du}{|Du|}) = u$, SIAM J. Math. Anal., 36 (2005), 1095-1129. doi: 10.1137/S0036141003430007.

[9]

K. A. Brakke, The Motion of a Surface by its Mean Curvature, Math. Notes, Princeton Univ. Press, Princeton, N. J., 1978.

[10]

H. Brézis, Opérateurs Maximaux Monotones et Semi-Groupes de Contractions dans les Espaces de Hilbert, North-Holland, 1973.

[11]

J. Buckland, Mean curvature flow with free boundary on smooth hypersurfaces, J. Reine Angew. Math., 586 (2005), 71-90. doi: 10.1515/crll.2005.2005.586.71.

[12]

V. Caselles, A. Chambolle and M. Novaga, The discontinuity set of solutions of the TV denoising problem and some extensions, Multiscale Model. Simul., 6 (2007), 879-894. doi: 10.1137/070683003.

[13]

V. Caselles, A. Chambolle and M. Novaga, Total variation in imaging, in Handbook of Mathematical Methods in Imaging, Springer, 2011, 1016-1057. doi: 10.1007/978-0-387-92920-0_23.

[14]

A. Cesaroni and M. Novaga, Long-time behavior of the mean curvature flow with periodic forcing, Comm. Partial Differential Equations, 38 (2013), 780-801. doi: 10.1080/03605302.2013.771508.

[15]

A. Chambolle, V. Caselles, D. Cremers, M. Novaga and T. Pock, An introduction to total variation for image analysis, in Theoretical Foundations and Numerical Methods for Sparse Recovery, Radon Series Comp. Appl. Math., {9}, Walter de Gruyter, Berlin, 2010, 263-340. doi: 10.1515/9783110226157.263.

[16]

K. Ecker, Estimates for evolutionary surfaces of prescribed mean curvature, Math. Z., 180 (1982), 179-192. doi: 10.1007/BF01318902.

[17]

K. Ecker and G. Huisken, Mean curvature evolution of entire graphs, Ann. of Math., 130 (1989), 453-471. doi: 10.2307/1971452.

[18]

K. Ecker and G. Huisken, Interior estimates for hypersurfaces moving by mean curvature, Invent. Math., 105 (1991), 547-569. doi: 10.1007/BF01232278.

[19]

M. Gage and R. Hamilton, The heat equation shrinking convex plane curves, J. Differential Geom., 23 (1986), 69-96.

[20]

C. Gerhardt, Evolutionary surfaces of prescribed mean curvature, J. Differential Equations, 36 (1980), 139-172. doi: 10.1016/0022-0396(80)90081-9.

[21]

E. Giusti, Minimal Surfaces and Functions of Bounded Variation, Monographs in Mathematics, Vol. 80, Boston-Basel-Stuttgart, Birkhäuser, 1984. doi: 10.1007/978-1-4684-9486-0.

[22]

A. Lichnewski and R. Temam, Pseudosolutions of the time-dependent minimal surface problem, J. Differential Equations, 30 (1978), 340-364. doi: 10.1016/0022-0396(78)90005-0.

[23]

A. Lunardi, Analytic Semigroups and Optimal Regularity in Parabolic Problems, Birkhäuser, Boston, 1995. doi: 10.1007/978-3-0348-9234-6.

[24]

U. Massari, Esistenza e regolarità delle ipersuperfici di curvatura media assegnata in $\mathbb{R}^{N}$, Arch. Ration. Mech. Anal., 55 (1974), 357-382.

[25]

I. Rudin, S. Osher and E. Fatemi, Nonlinear total variation based noise removal argorithms, Physica D, 60 (1992), 259-268. doi: 10.1016/0167-2789(92)90242-F.

[26]

A. Stahl, Regularity estimates for solutions to the mean curvature flow with a Neumann boundary condition, Calc. Var. Partial Differential Equations, 4 (1996), 385-407. doi: 10.1007/BF01190825.

[27]

I. Tamanini, Boundaries of Caccioppoli sets with Hölder-continuous normal vector, J. Reine Angew. Math., 334 (1982), 27-39. doi: 10.1515/crll.1982.334.27.

show all references

References:
[1]

L. Ambrosio, Corso Introduttivo alla Teoria Geometrica della Misura ed alle Superfici Minime, Edizioni della Scuola Normale, Pisa, 1997.

[2]

R. A. Adams, Sobolev Spaces, Academic Press, New York 1975.

[3]

F. Andreu, V. Caselles, J. I. Díaz and J. M. Mazón, Some qualitative properties for the total variation flow, J. Funct. Anal., 188 (2002), 516-547. doi: 10.1006/jfan.2001.3829.

[4]

F. Andreu, V. Caselles and J. M. Mazón, Parabolic Quasilinear Equations Minimizing Linear Growth Functionals, Oxford Mathematical Monographs, Birkhäuser, Basel, 2004. doi: 10.1007/978-3-0348-7928-6.

[5]

G. Anzellotti, Pairings between measures and bounded functions and compensated compactness, Ann. Mat. Pura Appl., 135 (1983), 293-318. doi: 10.1007/BF01781073.

[6]

G. Bellettini, Lecture Notes on Mean Curvature Flow, Barriers and Singular Perturbations, Edizioni della Scuola Normale, Pisa, 2013. doi: 10.1007/978-88-7642-429-8.

[7]

G. Bellettini, V. Caselles and M. Novaga, The total variation flow in $\mathbb{R}^N2$, J. Differential Equations, 184 (2002), 475-525. doi: 10.1006/jdeq.2001.4150.

[8]

G. Bellettini, V. Caselles and M. Novaga, Explicit solutions of the eigenvalue problem -div$(\frac{Du}{|Du|}) = u$, SIAM J. Math. Anal., 36 (2005), 1095-1129. doi: 10.1137/S0036141003430007.

[9]

K. A. Brakke, The Motion of a Surface by its Mean Curvature, Math. Notes, Princeton Univ. Press, Princeton, N. J., 1978.

[10]

H. Brézis, Opérateurs Maximaux Monotones et Semi-Groupes de Contractions dans les Espaces de Hilbert, North-Holland, 1973.

[11]

J. Buckland, Mean curvature flow with free boundary on smooth hypersurfaces, J. Reine Angew. Math., 586 (2005), 71-90. doi: 10.1515/crll.2005.2005.586.71.

[12]

V. Caselles, A. Chambolle and M. Novaga, The discontinuity set of solutions of the TV denoising problem and some extensions, Multiscale Model. Simul., 6 (2007), 879-894. doi: 10.1137/070683003.

[13]

V. Caselles, A. Chambolle and M. Novaga, Total variation in imaging, in Handbook of Mathematical Methods in Imaging, Springer, 2011, 1016-1057. doi: 10.1007/978-0-387-92920-0_23.

[14]

A. Cesaroni and M. Novaga, Long-time behavior of the mean curvature flow with periodic forcing, Comm. Partial Differential Equations, 38 (2013), 780-801. doi: 10.1080/03605302.2013.771508.

[15]

A. Chambolle, V. Caselles, D. Cremers, M. Novaga and T. Pock, An introduction to total variation for image analysis, in Theoretical Foundations and Numerical Methods for Sparse Recovery, Radon Series Comp. Appl. Math., {9}, Walter de Gruyter, Berlin, 2010, 263-340. doi: 10.1515/9783110226157.263.

[16]

K. Ecker, Estimates for evolutionary surfaces of prescribed mean curvature, Math. Z., 180 (1982), 179-192. doi: 10.1007/BF01318902.

[17]

K. Ecker and G. Huisken, Mean curvature evolution of entire graphs, Ann. of Math., 130 (1989), 453-471. doi: 10.2307/1971452.

[18]

K. Ecker and G. Huisken, Interior estimates for hypersurfaces moving by mean curvature, Invent. Math., 105 (1991), 547-569. doi: 10.1007/BF01232278.

[19]

M. Gage and R. Hamilton, The heat equation shrinking convex plane curves, J. Differential Geom., 23 (1986), 69-96.

[20]

C. Gerhardt, Evolutionary surfaces of prescribed mean curvature, J. Differential Equations, 36 (1980), 139-172. doi: 10.1016/0022-0396(80)90081-9.

[21]

E. Giusti, Minimal Surfaces and Functions of Bounded Variation, Monographs in Mathematics, Vol. 80, Boston-Basel-Stuttgart, Birkhäuser, 1984. doi: 10.1007/978-1-4684-9486-0.

[22]

A. Lichnewski and R. Temam, Pseudosolutions of the time-dependent minimal surface problem, J. Differential Equations, 30 (1978), 340-364. doi: 10.1016/0022-0396(78)90005-0.

[23]

A. Lunardi, Analytic Semigroups and Optimal Regularity in Parabolic Problems, Birkhäuser, Boston, 1995. doi: 10.1007/978-3-0348-9234-6.

[24]

U. Massari, Esistenza e regolarità delle ipersuperfici di curvatura media assegnata in $\mathbb{R}^{N}$, Arch. Ration. Mech. Anal., 55 (1974), 357-382.

[25]

I. Rudin, S. Osher and E. Fatemi, Nonlinear total variation based noise removal argorithms, Physica D, 60 (1992), 259-268. doi: 10.1016/0167-2789(92)90242-F.

[26]

A. Stahl, Regularity estimates for solutions to the mean curvature flow with a Neumann boundary condition, Calc. Var. Partial Differential Equations, 4 (1996), 385-407. doi: 10.1007/BF01190825.

[27]

I. Tamanini, Boundaries of Caccioppoli sets with Hölder-continuous normal vector, J. Reine Angew. Math., 334 (1982), 27-39. doi: 10.1515/crll.1982.334.27.

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