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On the classical limit of the Schrödinger equation
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 |
References:
[1] |
L. Ambrosio, Corso Introduttivo alla Teoria Geometrica della Misura ed alle Superfici Minime, Edizioni della Scuola Normale, Pisa, 1997. |
[2] | |
[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] | |
[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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