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Estimates of the derivatives of minimizers of a special class of variational integrals
Gamma-convergence of gradient flows on Hilbert and metric spaces and applications
1. | UPMC Univ Paris 06, UMR 7598 Laboratoire Jacques-Louis Lions, Paris, F-75005 |
References:
[1] |
N. Alikakos, P. Bates and X. Chen, Convergence of the Cahn-Hilliard equation to the Hele-Shaw model,, Arch. Rational Mech. Anal., 128 (1994), 165.
doi: 10.1007/BF00375025. |
[2] |
L. Ambrosio, Minimizing movements,, Rend. Accad. Naz. Sci, 19 (1995), 191.
|
[3] |
L. Ambrosio, N. Gigli and G. Savaré, "Gradient Flows in Metric Spaces and in The Space of Probability Measures,", Second edition, (2008).
|
[4] |
L. Ambrosio, E. Mainini and S. Serfaty, Gradient flow of the Chapman-Rubinstein-Schatzman model for signed vortices,, Ann. Institut Henri Poincaré Anal. Non Linéaire, 28 (2011), 217.
|
[5] |
L. Ambrosio and S. Serfaty, A gradient flow approach to an evolution problem arising in superconductivity,, Comm. Pure Appl. Math., 61 (2008), 1495.
doi: 10.1002/cpa.20223. |
[6] |
F. Bethuel, H. Brezis and F. Hélein, "Ginzburg-Landau Vortices,", Progress in Nonlinear Differential Equations and their Applications, 13 (1994).
|
[7] |
F. Bethuel, G. Orlandi and D. Smets, Convergence of the parabolic Ginzburg-Landau equation to motion by mean curvature,, Ann. of Math. (2), 163 (2006), 37.
doi: 10.4007/annals.2006.163.37. |
[8] |
F. Bethuel, G. Orlandi and D. Smets, Collisions and phase-vortex interactions in dissipative Ginzburg-Landau dynamics,, Duke Math. J., 130 (2005), 523.
doi: 10.1215/S0012-7094-05-13034-4. |
[9] |
F. Bethuel, G. Orlandi and D. Smets, Quantization and motion law for Ginzburg-Landau vortices,, Arch. Ration. Mech. Anal., 183 (2007), 315.
doi: 10.1007/s00205-006-0018-4. |
[10] |
F. Bethuel, G. Orlandi and D. Smets, Dynamics of multiple degree Ginzburg-Landau vortices,, omm. Math. Phys., 272 (2007), 229.
doi: 10.1007/s00220-007-0206-6. |
[11] |
L. Bronsard and R. V. Kohn, Motion by mean curvature as the singular limit of Ginzburg-Landau dynamics,, J. Differential Equations, 90 (1991), 211.
|
[12] |
S. J. Chapman, J. Rubinstein and M. Schatzman, A mean-field model of superconducting vortices,, European J. Appl. Math., 7 (1996), 97.
doi: 10.1017/S0956792500002242. |
[13] |
X. Chen, Generation and propagation of interfaces for reaction-diffusion equations,, J. Differential Equations, 96 (1992), 116.
doi: 10.1016/0022-0396(92)90146-E. |
[14] |
E. De Giorgi, New problems in $\Gamma$-convergence and $G$-convergence,, in, (1980), 183.
|
[15] |
E. De Giorgi, "New Problems on Minimizing Movements. Boundary Value Problems for Partial Differential Equations and Applications,", 81-98, 29 (1993), 81.
|
[16] |
E. De Giorgi, A. Marino and M. Tosques, Problems of evolution in metric spaces and maximal decreasing curve,, Att Accad Naz. Lincei Rend. Cl. Sci. Fis. Mat. Natur. (8), 68 (1980), 180.
|
[17] |
M. Degiovanni, A. Marino and M. Tosques, Evolution equations with lack of convexity,, Nonlinear Anal., 9 (1985), 1401.
doi: 10.1016/0362-546X(85)90098-7. |
[18] |
P. de Mottoni and M. Schatzman, Development of interfaces in $R^N$,, Proc. Roy. Soc. Edinburgh Sect. A, 116 (1990), 207.
|
[19] |
W. E, W. Ren and E. Vanden-Eijnden, Minimum action method for the study of rare events,, Comm. Pure Appl. Math., 57 (2004), 637.
|
[20] |
L. C. Evans, H. M. Soner and P. E. Souganidis, Phase transitions and generalized motion by mean curvature,, Comm. Pure Appl. Math, 45 (1992), 1097.
doi: 10.1002/cpa.3160450903. |
[21] |
J. Hutchinson and Y. Tonegawa, Convergence of phase interfaces in the van der Waals-Cahn-Hilliard theory,, Calc. Var. Partial Differential Equations, 10 (2000), 49.
|
[22] |
T. Ilmanen, Convergence of the Allen-Cahn equation to Brakke's motion by mean curvature,, J. Differential Geom, 38 (1993), 417.
|
[23] |
R. L. Jerrard, Vortex dynamics for the Ginzburg-Landau wave equation,, Calc. Var. Partial Differential Equations, 9 (1999), 1.
|
[24] |
R. L. Jerrard and H. M. Soner, Dynamics of Ginzburg-Landau vortices,, Arch. Rational Mech. Anal., 142 (1998), 99.
doi: 10.1007/s002050050085. |
[25] |
R. L. Jerrard and P. Sternberg, Critical points via Gamma-convergence: General theory and applications,, Jour. Eur. Math. Soc., 11 (2009), 705.
doi: 10.4171/JEMS/164. |
[26] |
H. Jian, A relation between $\Gamma$-convergence of functionals and their associated gradient flows,, Sci. China Ser. A, 42 (1999), 133.
doi: 10.1007/BF02876564. |
[27] |
R. V. Kohn, F. Otto, M. G. Reznikoff and E. Vanden-Eijnden, Action minimization and sharp-interface limits for the stochastic Allen-Cahn equation,, Comm. Pure Appl. Math., 60 (2007), 393.
doi: 10.1002/cpa.20144. |
[28] |
R. V. Kohn, M. G. Reznikoff and Y. Tonegawa, Sharp-interface limit of the Allen-Cahn action functional in one space dimension,, Calc. Var. Partial Differential Equations, 25 (2006), 503.
|
[29] |
M. Kurzke, The gradient flow motion of boundary vortices,, Ann. Inst. H. Poincaré Anal. Non Linéaire, 24 (2007), 91.
|
[30] |
N. Le, A Gamma-Convergence approach to the Cahn-Hilliard equation,, Calc. Var. Partial Differential Equations, 32 (2008), 499.
|
[31] |
N. Le, On the convergence of the Ohta-Kawasaki Equation to motion by nonlocal Mullins-Sekerka Law,, SIAM. J. Math. Analysis., 42 (2010), 1602.
doi: 10.1137/090768643. |
[32] |
F. H. Lin, Some dynamical properties of Ginzburg-Landau vortices,, Comm. Pure Appl. Math., 49 (1996), 323.
|
[33] |
A. Marino, C. Saccon and M. Tosques, Curves of maximal slope and parabolic variational inequalities on nonconvex constraints,, Ann Scuola Norm. Sup. Pisa Cl. Sci. (4), 16 (1989), 281.
|
[34] |
A. Mielke, Weak convergence methods for Hamiltonian multiscale problems,, Discrete Contin. Dyn. Syst. Ser. A, 20 (2008), 53.
doi: 10.3934/dcds.2008.20.53. |
[35] |
A. Mielke, T. Roubi\vcek and U. Stefanelli, $\Gamma$-limits and relaxations for rate-independent evolutionary problems,, Calc. Var. PDE, 31 (2008), 387.
doi: 10.1007/s00526-007-0119-4. |
[36] |
L. Modica and S. Mortola, Il limite nella $\Gamma $-convergenza di una famiglia di funzionali ellittici,, Boll. Un. Mat. Ital. A (5), 14 (1977), 526.
|
[37] |
L. Mugnai and M. Röger, The Allen-Cahn action functional in higher dimensions,, Interfaces Free Bound., 10 (2008), 45.
doi: 10.4171/IFB/179. |
[38] |
L. Mugnai and M. Röger, Convergence of the perturbed Allen-Cahn equations to forced mean curvature flow,, preprint., (). Google Scholar |
[39] |
M. Novaga and E. Valdinocci, Closed curves of prescribed curvature and a pinning effect,, Networks Heterog. Media \textbf{6} (2011), 6 (2011), 77.
|
[40] |
C. Ortner, "Two Variational Techniques for the Approximation of Curves of Maximal Slope,", Technical report NA05/10, (2005). Google Scholar |
[41] |
C. Ortner, Gradient flows as a selection procedure for equilibria of nonconvex energies,, SIAM J. Math. Anal., 38 (2006), 1214.
doi: 10.1137/050643982. |
[42] |
R. L. Pego, Front migration in the nonlinear Cahn-Hilliard equation,, Proc. Roy. Soc. London Ser. A, 422 (1989), 261.
doi: 10.1098/rspa.1989.0027. |
[43] |
M. Röger and R. Schätzle, On a modified conjecture of De Giorgi,, Math. Z., 254 (2006), 675.
doi: 10.1007/s00209-006-0002-6. |
[44] |
E. Sandier and S. Serfaty, Gamma-convergence of gradient flows with applications to Ginzburg-Landau,, Comm. Pure Appl. Math, 57 (2004), 1627.
doi: 10.1002/cpa.20046. |
[45] |
E. Sandier and S. Serfaty, Limiting vorticities for the Ginzburg-Landau equations,, Duke Math J., 117 (2003), 403.
doi: 10.1215/S0012-7094-03-11732-9. |
[46] |
E. Sandier and S. Serfaty, A product-estimate for Ginzburg-Landau and corollaries,, J. Func. Anal., 211 (2004), 219.
doi: 10.1016/S0022-1236(03)00199-X. |
[47] |
E. Sandier and S. Serfaty, "Vortices in the Magnetic Ginzburg-Landau Model,", Progress in Nonlinear Differential Equations and their Applications, 70 (2007).
|
[48] |
E. Sandier and S. Serfaty, A rigorous derivation of a free-boundary problem arising in superconductivity,, Annales Scientifiques de l'ENS (4), 33 (2000), 561.
|
[49] |
N. Sato, A simple proof of convergence of the Allen-Cahn Equation to Brakke's motion by mean curvature,, Indiana Univ. Math. J, 57 (2008), 1743.
doi: 10.1512/iumj.2008.57.3283. |
[50] |
, R. Schätzle,, private communication., (). Google Scholar |
[51] |
S. Serfaty, Stability in 2D Ginzburg-Landau passes to the limit,, Indiana Univ. Math. J., 54 (2005), 199.
doi: 10.1512/iumj.2005.54.2497. |
[52] |
S. Serfaty, Vortex collisions and energy-dissipation rates in the Ginzburg-Landau heat flow. Part I: Study of the perturbed Ginzburg-Landau equation,, Journal Eur. Math Society, 9 (2007), 177.
doi: 10.4171/JEMS/84. |
[53] |
C. Villani, "Optimal Transport. Old and New,", Grundlehren der Mathematischen Wissenschaften, 338 (2009).
|
[54] |
M. G. Westdickenberg and Y. Tonegawa, Higher multiplicity in the one-dimensional Allen-Cahn action functional,, Indiana Univ. Math. J., 56 (2007), 2935.
doi: 10.1512/iumj.2007.56.3182. |
show all references
References:
[1] |
N. Alikakos, P. Bates and X. Chen, Convergence of the Cahn-Hilliard equation to the Hele-Shaw model,, Arch. Rational Mech. Anal., 128 (1994), 165.
doi: 10.1007/BF00375025. |
[2] |
L. Ambrosio, Minimizing movements,, Rend. Accad. Naz. Sci, 19 (1995), 191.
|
[3] |
L. Ambrosio, N. Gigli and G. Savaré, "Gradient Flows in Metric Spaces and in The Space of Probability Measures,", Second edition, (2008).
|
[4] |
L. Ambrosio, E. Mainini and S. Serfaty, Gradient flow of the Chapman-Rubinstein-Schatzman model for signed vortices,, Ann. Institut Henri Poincaré Anal. Non Linéaire, 28 (2011), 217.
|
[5] |
L. Ambrosio and S. Serfaty, A gradient flow approach to an evolution problem arising in superconductivity,, Comm. Pure Appl. Math., 61 (2008), 1495.
doi: 10.1002/cpa.20223. |
[6] |
F. Bethuel, H. Brezis and F. Hélein, "Ginzburg-Landau Vortices,", Progress in Nonlinear Differential Equations and their Applications, 13 (1994).
|
[7] |
F. Bethuel, G. Orlandi and D. Smets, Convergence of the parabolic Ginzburg-Landau equation to motion by mean curvature,, Ann. of Math. (2), 163 (2006), 37.
doi: 10.4007/annals.2006.163.37. |
[8] |
F. Bethuel, G. Orlandi and D. Smets, Collisions and phase-vortex interactions in dissipative Ginzburg-Landau dynamics,, Duke Math. J., 130 (2005), 523.
doi: 10.1215/S0012-7094-05-13034-4. |
[9] |
F. Bethuel, G. Orlandi and D. Smets, Quantization and motion law for Ginzburg-Landau vortices,, Arch. Ration. Mech. Anal., 183 (2007), 315.
doi: 10.1007/s00205-006-0018-4. |
[10] |
F. Bethuel, G. Orlandi and D. Smets, Dynamics of multiple degree Ginzburg-Landau vortices,, omm. Math. Phys., 272 (2007), 229.
doi: 10.1007/s00220-007-0206-6. |
[11] |
L. Bronsard and R. V. Kohn, Motion by mean curvature as the singular limit of Ginzburg-Landau dynamics,, J. Differential Equations, 90 (1991), 211.
|
[12] |
S. J. Chapman, J. Rubinstein and M. Schatzman, A mean-field model of superconducting vortices,, European J. Appl. Math., 7 (1996), 97.
doi: 10.1017/S0956792500002242. |
[13] |
X. Chen, Generation and propagation of interfaces for reaction-diffusion equations,, J. Differential Equations, 96 (1992), 116.
doi: 10.1016/0022-0396(92)90146-E. |
[14] |
E. De Giorgi, New problems in $\Gamma$-convergence and $G$-convergence,, in, (1980), 183.
|
[15] |
E. De Giorgi, "New Problems on Minimizing Movements. Boundary Value Problems for Partial Differential Equations and Applications,", 81-98, 29 (1993), 81.
|
[16] |
E. De Giorgi, A. Marino and M. Tosques, Problems of evolution in metric spaces and maximal decreasing curve,, Att Accad Naz. Lincei Rend. Cl. Sci. Fis. Mat. Natur. (8), 68 (1980), 180.
|
[17] |
M. Degiovanni, A. Marino and M. Tosques, Evolution equations with lack of convexity,, Nonlinear Anal., 9 (1985), 1401.
doi: 10.1016/0362-546X(85)90098-7. |
[18] |
P. de Mottoni and M. Schatzman, Development of interfaces in $R^N$,, Proc. Roy. Soc. Edinburgh Sect. A, 116 (1990), 207.
|
[19] |
W. E, W. Ren and E. Vanden-Eijnden, Minimum action method for the study of rare events,, Comm. Pure Appl. Math., 57 (2004), 637.
|
[20] |
L. C. Evans, H. M. Soner and P. E. Souganidis, Phase transitions and generalized motion by mean curvature,, Comm. Pure Appl. Math, 45 (1992), 1097.
doi: 10.1002/cpa.3160450903. |
[21] |
J. Hutchinson and Y. Tonegawa, Convergence of phase interfaces in the van der Waals-Cahn-Hilliard theory,, Calc. Var. Partial Differential Equations, 10 (2000), 49.
|
[22] |
T. Ilmanen, Convergence of the Allen-Cahn equation to Brakke's motion by mean curvature,, J. Differential Geom, 38 (1993), 417.
|
[23] |
R. L. Jerrard, Vortex dynamics for the Ginzburg-Landau wave equation,, Calc. Var. Partial Differential Equations, 9 (1999), 1.
|
[24] |
R. L. Jerrard and H. M. Soner, Dynamics of Ginzburg-Landau vortices,, Arch. Rational Mech. Anal., 142 (1998), 99.
doi: 10.1007/s002050050085. |
[25] |
R. L. Jerrard and P. Sternberg, Critical points via Gamma-convergence: General theory and applications,, Jour. Eur. Math. Soc., 11 (2009), 705.
doi: 10.4171/JEMS/164. |
[26] |
H. Jian, A relation between $\Gamma$-convergence of functionals and their associated gradient flows,, Sci. China Ser. A, 42 (1999), 133.
doi: 10.1007/BF02876564. |
[27] |
R. V. Kohn, F. Otto, M. G. Reznikoff and E. Vanden-Eijnden, Action minimization and sharp-interface limits for the stochastic Allen-Cahn equation,, Comm. Pure Appl. Math., 60 (2007), 393.
doi: 10.1002/cpa.20144. |
[28] |
R. V. Kohn, M. G. Reznikoff and Y. Tonegawa, Sharp-interface limit of the Allen-Cahn action functional in one space dimension,, Calc. Var. Partial Differential Equations, 25 (2006), 503.
|
[29] |
M. Kurzke, The gradient flow motion of boundary vortices,, Ann. Inst. H. Poincaré Anal. Non Linéaire, 24 (2007), 91.
|
[30] |
N. Le, A Gamma-Convergence approach to the Cahn-Hilliard equation,, Calc. Var. Partial Differential Equations, 32 (2008), 499.
|
[31] |
N. Le, On the convergence of the Ohta-Kawasaki Equation to motion by nonlocal Mullins-Sekerka Law,, SIAM. J. Math. Analysis., 42 (2010), 1602.
doi: 10.1137/090768643. |
[32] |
F. H. Lin, Some dynamical properties of Ginzburg-Landau vortices,, Comm. Pure Appl. Math., 49 (1996), 323.
|
[33] |
A. Marino, C. Saccon and M. Tosques, Curves of maximal slope and parabolic variational inequalities on nonconvex constraints,, Ann Scuola Norm. Sup. Pisa Cl. Sci. (4), 16 (1989), 281.
|
[34] |
A. Mielke, Weak convergence methods for Hamiltonian multiscale problems,, Discrete Contin. Dyn. Syst. Ser. A, 20 (2008), 53.
doi: 10.3934/dcds.2008.20.53. |
[35] |
A. Mielke, T. Roubi\vcek and U. Stefanelli, $\Gamma$-limits and relaxations for rate-independent evolutionary problems,, Calc. Var. PDE, 31 (2008), 387.
doi: 10.1007/s00526-007-0119-4. |
[36] |
L. Modica and S. Mortola, Il limite nella $\Gamma $-convergenza di una famiglia di funzionali ellittici,, Boll. Un. Mat. Ital. A (5), 14 (1977), 526.
|
[37] |
L. Mugnai and M. Röger, The Allen-Cahn action functional in higher dimensions,, Interfaces Free Bound., 10 (2008), 45.
doi: 10.4171/IFB/179. |
[38] |
L. Mugnai and M. Röger, Convergence of the perturbed Allen-Cahn equations to forced mean curvature flow,, preprint., (). Google Scholar |
[39] |
M. Novaga and E. Valdinocci, Closed curves of prescribed curvature and a pinning effect,, Networks Heterog. Media \textbf{6} (2011), 6 (2011), 77.
|
[40] |
C. Ortner, "Two Variational Techniques for the Approximation of Curves of Maximal Slope,", Technical report NA05/10, (2005). Google Scholar |
[41] |
C. Ortner, Gradient flows as a selection procedure for equilibria of nonconvex energies,, SIAM J. Math. Anal., 38 (2006), 1214.
doi: 10.1137/050643982. |
[42] |
R. L. Pego, Front migration in the nonlinear Cahn-Hilliard equation,, Proc. Roy. Soc. London Ser. A, 422 (1989), 261.
doi: 10.1098/rspa.1989.0027. |
[43] |
M. Röger and R. Schätzle, On a modified conjecture of De Giorgi,, Math. Z., 254 (2006), 675.
doi: 10.1007/s00209-006-0002-6. |
[44] |
E. Sandier and S. Serfaty, Gamma-convergence of gradient flows with applications to Ginzburg-Landau,, Comm. Pure Appl. Math, 57 (2004), 1627.
doi: 10.1002/cpa.20046. |
[45] |
E. Sandier and S. Serfaty, Limiting vorticities for the Ginzburg-Landau equations,, Duke Math J., 117 (2003), 403.
doi: 10.1215/S0012-7094-03-11732-9. |
[46] |
E. Sandier and S. Serfaty, A product-estimate for Ginzburg-Landau and corollaries,, J. Func. Anal., 211 (2004), 219.
doi: 10.1016/S0022-1236(03)00199-X. |
[47] |
E. Sandier and S. Serfaty, "Vortices in the Magnetic Ginzburg-Landau Model,", Progress in Nonlinear Differential Equations and their Applications, 70 (2007).
|
[48] |
E. Sandier and S. Serfaty, A rigorous derivation of a free-boundary problem arising in superconductivity,, Annales Scientifiques de l'ENS (4), 33 (2000), 561.
|
[49] |
N. Sato, A simple proof of convergence of the Allen-Cahn Equation to Brakke's motion by mean curvature,, Indiana Univ. Math. J, 57 (2008), 1743.
doi: 10.1512/iumj.2008.57.3283. |
[50] |
, R. Schätzle,, private communication., (). Google Scholar |
[51] |
S. Serfaty, Stability in 2D Ginzburg-Landau passes to the limit,, Indiana Univ. Math. J., 54 (2005), 199.
doi: 10.1512/iumj.2005.54.2497. |
[52] |
S. Serfaty, Vortex collisions and energy-dissipation rates in the Ginzburg-Landau heat flow. Part I: Study of the perturbed Ginzburg-Landau equation,, Journal Eur. Math Society, 9 (2007), 177.
doi: 10.4171/JEMS/84. |
[53] |
C. Villani, "Optimal Transport. Old and New,", Grundlehren der Mathematischen Wissenschaften, 338 (2009).
|
[54] |
M. G. Westdickenberg and Y. Tonegawa, Higher multiplicity in the one-dimensional Allen-Cahn action functional,, Indiana Univ. Math. J., 56 (2007), 2935.
doi: 10.1512/iumj.2007.56.3182. |
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