Eventual regularity for the parabolic minimal surface equation

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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

Abstract
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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 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.

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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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