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

Eventual regularity for the parabolic minimal surface equation

Giovanni Bellettini 1, , Matteo Novaga 2, and  Giandomenico Orlandi 3,

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.
Keywords: eventual regularity of solutions., Parabolic minimal surface equation.
Mathematics Subject Classification: Primary: 35K55, 47H20; Secondary: 35K9.
Citation: Giovanni Bellettini, Matteo Novaga, Giandomenico Orlandi. Eventual regularity for the parabolic minimal surface equation. Discrete & 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.  Google Scholar

[2]

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

[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.  Google Scholar

[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.  Google Scholar

[5]

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

[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.  Google Scholar

[7]

G. Bellettini, V. Caselles and M. Novaga, The total variation flow in $\mathbbR^N$, J. Differential Equations, 184 (2002), 475-525. doi: 10.1006/jdeq.2001.4150.  Google Scholar

[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.  Google Scholar

[9]

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

[10]

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

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[16]

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

[17]

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

[18]

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

[19]

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

[20]

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

[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.  Google Scholar

[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.  Google Scholar

[23]

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

[24]

U. Massari, Esistenza e regolarità delle ipersuperfici di curvatura media assegnata in $\mathbbR^n$, Arch. Ration. Mech. Anal., 55 (1974), 357-382.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

show all references

References:
[1]

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

[2]

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

[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.  Google Scholar

[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.  Google Scholar

[5]

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

[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.  Google Scholar

[7]

G. Bellettini, V. Caselles and M. Novaga, The total variation flow in $\mathbbR^N$, J. Differential Equations, 184 (2002), 475-525. doi: 10.1006/jdeq.2001.4150.  Google Scholar

[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.  Google Scholar

[9]

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

[10]

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

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[16]

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

[17]

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

[18]

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

[19]

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

[20]

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

[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.  Google Scholar

[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.  Google Scholar

[23]

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

[24]

U. Massari, Esistenza e regolarità delle ipersuperfici di curvatura media assegnata in $\mathbbR^n$, Arch. Ration. Mech. Anal., 55 (1974), 357-382.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

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