December  2011, 31(4): 1427-1451. doi: 10.3934/dcds.2011.31.1427

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

Received  May 2009 Revised  July 2010 Published  September 2011

We are concerned with $\Gamma$-convergence of gradient flows, which is a notion meant to ensure that if a family of energy functionals depending of a parameter $\Gamma$-converges, then the solutions to the associated gradient flows converge as well. In this paper we present both a review of the abstract "theory" and of the applications it has had, and a generalization of the scheme to metric spaces which has not appeared elsewhere. We also mention open problems and perspectives.
Citation: Sylvia Serfaty. Gamma-convergence of gradient flows on Hilbert and metric spaces and applications. Discrete & Continuous Dynamical Systems, 2011, 31 (4) : 1427-1451. doi: 10.3934/dcds.2011.31.1427
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), 165-205. doi: 10.1007/BF00375025.  Google Scholar

[2]

L. Ambrosio, Minimizing movements, Rend. Accad. Naz. Sci, XL Mem. Mat. Appl. (5), 19 (1995), 191-246.  Google Scholar

[3]

L. Ambrosio, N. Gigli and G. Savaré, "Gradient Flows in Metric Spaces and in The Space of Probability Measures," Second edition, Lectures in Mathematics ETH Zürich, Birkhäuser Verlag, Basel, 2008.  Google Scholar

[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), 217-246.  Google Scholar

[5]

L. Ambrosio and S. Serfaty, A gradient flow approach to an evolution problem arising in superconductivity, Comm. Pure Appl. Math., 61 (2008), 1495-1539. doi: 10.1002/cpa.20223.  Google Scholar

[6]

F. Bethuel, H. Brezis and F. Hélein, "Ginzburg-Landau Vortices," Progress in Nonlinear Differential Equations and their Applications, 13, Birkhäuser Boston, Inc., Boston, MA, 1994.  Google Scholar

[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), 37-163. doi: 10.4007/annals.2006.163.37.  Google Scholar

[8]

F. Bethuel, G. Orlandi and D. Smets, Collisions and phase-vortex interactions in dissipative Ginzburg-Landau dynamics, Duke Math. J., 130 (2005), 523-614. doi: 10.1215/S0012-7094-05-13034-4.  Google Scholar

[9]

F. Bethuel, G. Orlandi and D. Smets, Quantization and motion law for Ginzburg-Landau vortices, Arch. Ration. Mech. Anal., 183 (2007), 315-370. doi: 10.1007/s00205-006-0018-4.  Google Scholar

[10]

F. Bethuel, G. Orlandi and D. Smets, Dynamics of multiple degree Ginzburg-Landau vortices, omm. Math. Phys., 272 (2007), 229-261. doi: 10.1007/s00220-007-0206-6.  Google Scholar

[11]

L. Bronsard and R. V. Kohn, Motion by mean curvature as the singular limit of Ginzburg-Landau dynamics, J. Differential Equations, 90 (1991), 211-237.  Google Scholar

[12]

S. J. Chapman, J. Rubinstein and M. Schatzman, A mean-field model of superconducting vortices, European J. Appl. Math., 7 (1996), 97-111. doi: 10.1017/S0956792500002242.  Google Scholar

[13]

X. Chen, Generation and propagation of interfaces for reaction-diffusion equations, J. Differential Equations, 96 (1992), 116-141. doi: 10.1016/0022-0396(92)90146-E.  Google Scholar

[14]

E. De Giorgi, New problems in $\Gamma$-convergence and $G$-convergence, in "Free Boundary Problems, Vol. II" (Pavia, 1979), Ist. Naz. Alta Mat. Francesco Severi, Rome, (1980), 183-194.  Google Scholar

[15]

E. De Giorgi, "New Problems on Minimizing Movements. Boundary Value Problems for Partial Differential Equations and Applications," 81-98, RMA Res. Notes Appl. Math., 29, Masson, Paris, 1993.  Google Scholar

[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), 180-187.  Google Scholar

[17]

M. Degiovanni, A. Marino and M. Tosques, Evolution equations with lack of convexity, Nonlinear Anal., 9 (1985), 1401-1443. doi: 10.1016/0362-546X(85)90098-7.  Google Scholar

[18]

P. de Mottoni and M. Schatzman, Development of interfaces in $R^N$, Proc. Roy. Soc. Edinburgh Sect. A, 116 (1990), 207-220.  Google Scholar

[19]

W. E, W. Ren and E. Vanden-Eijnden, Minimum action method for the study of rare events, Comm. Pure Appl. Math., 57 (2004), 637-656.  Google Scholar

[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), 1097-1123. doi: 10.1002/cpa.3160450903.  Google Scholar

[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), 49-84.  Google Scholar

[22]

T. Ilmanen, Convergence of the Allen-Cahn equation to Brakke's motion by mean curvature, J. Differential Geom, 38 (1993), 417-461.  Google Scholar

[23]

R. L. Jerrard, Vortex dynamics for the Ginzburg-Landau wave equation, Calc. Var. Partial Differential Equations, 9 (1999), 1-30.  Google Scholar

[24]

R. L. Jerrard and H. M. Soner, Dynamics of Ginzburg-Landau vortices, Arch. Rational Mech. Anal., 142 (1998), 99-125. doi: 10.1007/s002050050085.  Google Scholar

[25]

R. L. Jerrard and P. Sternberg, Critical points via Gamma-convergence: General theory and applications, Jour. Eur. Math. Soc., 11 (2009), 705-753. doi: 10.4171/JEMS/164.  Google Scholar

[26]

H. Jian, A relation between $\Gamma$-convergence of functionals and their associated gradient flows, Sci. China Ser. A, 42 (1999), 133-139. doi: 10.1007/BF02876564.  Google Scholar

[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), 393-438. doi: 10.1002/cpa.20144.  Google Scholar

[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), 503-534.  Google Scholar

[29]

M. Kurzke, The gradient flow motion of boundary vortices, Ann. Inst. H. Poincaré Anal. Non Linéaire, 24 (2007), 91-112.  Google Scholar

[30]

N. Le, A Gamma-Convergence approach to the Cahn-Hilliard equation, Calc. Var. Partial Differential Equations, 32 (2008), 499-522.  Google Scholar

[31]

N. Le, On the convergence of the Ohta-Kawasaki Equation to motion by nonlocal Mullins-Sekerka Law, SIAM. J. Math. Analysis., 42 (2010), 1602-1638. doi: 10.1137/090768643.  Google Scholar

[32]

F. H. Lin, Some dynamical properties of Ginzburg-Landau vortices, Comm. Pure Appl. Math., 49 (1996), 323-359.  Google Scholar

[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), 281-330.  Google Scholar

[34]

A. Mielke, Weak convergence methods for Hamiltonian multiscale problems, Discrete Contin. Dyn. Syst. Ser. A, 20 (2008), 53-79. doi: 10.3934/dcds.2008.20.53.  Google Scholar

[35]

A. Mielke, T. Roubi\vcek and U. Stefanelli, $\Gamma$-limits and relaxations for rate-independent evolutionary problems, Calc. Var. PDE, 31 (2008), 387-416. doi: 10.1007/s00526-007-0119-4.  Google Scholar

[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), 526-529.  Google Scholar

[37]

L. Mugnai and M. Röger, The Allen-Cahn action functional in higher dimensions, Interfaces Free Bound., 10 (2008), 45-78. doi: 10.4171/IFB/179.  Google Scholar

[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 6 (2011), 77-88.  Google Scholar

[40]

C. Ortner, "Two Variational Techniques for the Approximation of Curves of Maximal Slope," Technical report NA05/10, Oxford University Computing Laboratory, Oxford, UK, 2005. Google Scholar

[41]

C. Ortner, Gradient flows as a selection procedure for equilibria of nonconvex energies, SIAM J. Math. Anal., 38 (2006), 1214-1234. doi: 10.1137/050643982.  Google Scholar

[42]

R. L. Pego, Front migration in the nonlinear Cahn-Hilliard equation, Proc. Roy. Soc. London Ser. A, 422 (1989), 261-278. doi: 10.1098/rspa.1989.0027.  Google Scholar

[43]

M. Röger and R. Schätzle, On a modified conjecture of De Giorgi, Math. Z., 254 (2006), 675-714. doi: 10.1007/s00209-006-0002-6.  Google Scholar

[44]

E. Sandier and S. Serfaty, Gamma-convergence of gradient flows with applications to Ginzburg-Landau, Comm. Pure Appl. Math, 57 (2004), 1627-1672. doi: 10.1002/cpa.20046.  Google Scholar

[45]

E. Sandier and S. Serfaty, Limiting vorticities for the Ginzburg-Landau equations, Duke Math J., 117 (2003), 403-446. doi: 10.1215/S0012-7094-03-11732-9.  Google Scholar

[46]

E. Sandier and S. Serfaty, A product-estimate for Ginzburg-Landau and corollaries, J. Func. Anal., 211 (2004), 219-244. doi: 10.1016/S0022-1236(03)00199-X.  Google Scholar

[47]

E. Sandier and S. Serfaty, "Vortices in the Magnetic Ginzburg-Landau Model," Progress in Nonlinear Differential Equations and their Applications, 70, Birkhäuser Boston, Inc., Boston, MA, 2007.  Google Scholar

[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), 561-592.  Google Scholar

[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), 1743-1751. doi: 10.1512/iumj.2008.57.3283.  Google Scholar

[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), 199-221. doi: 10.1512/iumj.2005.54.2497.  Google Scholar

[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), 177-217, Part II: The dynamics, Journal Eur. Math Society, 9 (2007), 383-426. doi: 10.4171/JEMS/84.  Google Scholar

[53]

C. Villani, "Optimal Transport. Old and New," Grundlehren der Mathematischen Wissenschaften, 338, Springer-Verlag, Berlin, 2009.  Google Scholar

[54]

M. G. Westdickenberg and Y. Tonegawa, Higher multiplicity in the one-dimensional Allen-Cahn action functional, Indiana Univ. Math. J., 56 (2007), 2935-2989. doi: 10.1512/iumj.2007.56.3182.  Google Scholar

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), 165-205. doi: 10.1007/BF00375025.  Google Scholar

[2]

L. Ambrosio, Minimizing movements, Rend. Accad. Naz. Sci, XL Mem. Mat. Appl. (5), 19 (1995), 191-246.  Google Scholar

[3]

L. Ambrosio, N. Gigli and G. Savaré, "Gradient Flows in Metric Spaces and in The Space of Probability Measures," Second edition, Lectures in Mathematics ETH Zürich, Birkhäuser Verlag, Basel, 2008.  Google Scholar

[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), 217-246.  Google Scholar

[5]

L. Ambrosio and S. Serfaty, A gradient flow approach to an evolution problem arising in superconductivity, Comm. Pure Appl. Math., 61 (2008), 1495-1539. doi: 10.1002/cpa.20223.  Google Scholar

[6]

F. Bethuel, H. Brezis and F. Hélein, "Ginzburg-Landau Vortices," Progress in Nonlinear Differential Equations and their Applications, 13, Birkhäuser Boston, Inc., Boston, MA, 1994.  Google Scholar

[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), 37-163. doi: 10.4007/annals.2006.163.37.  Google Scholar

[8]

F. Bethuel, G. Orlandi and D. Smets, Collisions and phase-vortex interactions in dissipative Ginzburg-Landau dynamics, Duke Math. J., 130 (2005), 523-614. doi: 10.1215/S0012-7094-05-13034-4.  Google Scholar

[9]

F. Bethuel, G. Orlandi and D. Smets, Quantization and motion law for Ginzburg-Landau vortices, Arch. Ration. Mech. Anal., 183 (2007), 315-370. doi: 10.1007/s00205-006-0018-4.  Google Scholar

[10]

F. Bethuel, G. Orlandi and D. Smets, Dynamics of multiple degree Ginzburg-Landau vortices, omm. Math. Phys., 272 (2007), 229-261. doi: 10.1007/s00220-007-0206-6.  Google Scholar

[11]

L. Bronsard and R. V. Kohn, Motion by mean curvature as the singular limit of Ginzburg-Landau dynamics, J. Differential Equations, 90 (1991), 211-237.  Google Scholar

[12]

S. J. Chapman, J. Rubinstein and M. Schatzman, A mean-field model of superconducting vortices, European J. Appl. Math., 7 (1996), 97-111. doi: 10.1017/S0956792500002242.  Google Scholar

[13]

X. Chen, Generation and propagation of interfaces for reaction-diffusion equations, J. Differential Equations, 96 (1992), 116-141. doi: 10.1016/0022-0396(92)90146-E.  Google Scholar

[14]

E. De Giorgi, New problems in $\Gamma$-convergence and $G$-convergence, in "Free Boundary Problems, Vol. II" (Pavia, 1979), Ist. Naz. Alta Mat. Francesco Severi, Rome, (1980), 183-194.  Google Scholar

[15]

E. De Giorgi, "New Problems on Minimizing Movements. Boundary Value Problems for Partial Differential Equations and Applications," 81-98, RMA Res. Notes Appl. Math., 29, Masson, Paris, 1993.  Google Scholar

[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), 180-187.  Google Scholar

[17]

M. Degiovanni, A. Marino and M. Tosques, Evolution equations with lack of convexity, Nonlinear Anal., 9 (1985), 1401-1443. doi: 10.1016/0362-546X(85)90098-7.  Google Scholar

[18]

P. de Mottoni and M. Schatzman, Development of interfaces in $R^N$, Proc. Roy. Soc. Edinburgh Sect. A, 116 (1990), 207-220.  Google Scholar

[19]

W. E, W. Ren and E. Vanden-Eijnden, Minimum action method for the study of rare events, Comm. Pure Appl. Math., 57 (2004), 637-656.  Google Scholar

[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), 1097-1123. doi: 10.1002/cpa.3160450903.  Google Scholar

[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), 49-84.  Google Scholar

[22]

T. Ilmanen, Convergence of the Allen-Cahn equation to Brakke's motion by mean curvature, J. Differential Geom, 38 (1993), 417-461.  Google Scholar

[23]

R. L. Jerrard, Vortex dynamics for the Ginzburg-Landau wave equation, Calc. Var. Partial Differential Equations, 9 (1999), 1-30.  Google Scholar

[24]

R. L. Jerrard and H. M. Soner, Dynamics of Ginzburg-Landau vortices, Arch. Rational Mech. Anal., 142 (1998), 99-125. doi: 10.1007/s002050050085.  Google Scholar

[25]

R. L. Jerrard and P. Sternberg, Critical points via Gamma-convergence: General theory and applications, Jour. Eur. Math. Soc., 11 (2009), 705-753. doi: 10.4171/JEMS/164.  Google Scholar

[26]

H. Jian, A relation between $\Gamma$-convergence of functionals and their associated gradient flows, Sci. China Ser. A, 42 (1999), 133-139. doi: 10.1007/BF02876564.  Google Scholar

[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), 393-438. doi: 10.1002/cpa.20144.  Google Scholar

[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), 503-534.  Google Scholar

[29]

M. Kurzke, The gradient flow motion of boundary vortices, Ann. Inst. H. Poincaré Anal. Non Linéaire, 24 (2007), 91-112.  Google Scholar

[30]

N. Le, A Gamma-Convergence approach to the Cahn-Hilliard equation, Calc. Var. Partial Differential Equations, 32 (2008), 499-522.  Google Scholar

[31]

N. Le, On the convergence of the Ohta-Kawasaki Equation to motion by nonlocal Mullins-Sekerka Law, SIAM. J. Math. Analysis., 42 (2010), 1602-1638. doi: 10.1137/090768643.  Google Scholar

[32]

F. H. Lin, Some dynamical properties of Ginzburg-Landau vortices, Comm. Pure Appl. Math., 49 (1996), 323-359.  Google Scholar

[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), 281-330.  Google Scholar

[34]

A. Mielke, Weak convergence methods for Hamiltonian multiscale problems, Discrete Contin. Dyn. Syst. Ser. A, 20 (2008), 53-79. doi: 10.3934/dcds.2008.20.53.  Google Scholar

[35]

A. Mielke, T. Roubi\vcek and U. Stefanelli, $\Gamma$-limits and relaxations for rate-independent evolutionary problems, Calc. Var. PDE, 31 (2008), 387-416. doi: 10.1007/s00526-007-0119-4.  Google Scholar

[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), 526-529.  Google Scholar

[37]

L. Mugnai and M. Röger, The Allen-Cahn action functional in higher dimensions, Interfaces Free Bound., 10 (2008), 45-78. doi: 10.4171/IFB/179.  Google Scholar

[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 6 (2011), 77-88.  Google Scholar

[40]

C. Ortner, "Two Variational Techniques for the Approximation of Curves of Maximal Slope," Technical report NA05/10, Oxford University Computing Laboratory, Oxford, UK, 2005. Google Scholar

[41]

C. Ortner, Gradient flows as a selection procedure for equilibria of nonconvex energies, SIAM J. Math. Anal., 38 (2006), 1214-1234. doi: 10.1137/050643982.  Google Scholar

[42]

R. L. Pego, Front migration in the nonlinear Cahn-Hilliard equation, Proc. Roy. Soc. London Ser. A, 422 (1989), 261-278. doi: 10.1098/rspa.1989.0027.  Google Scholar

[43]

M. Röger and R. Schätzle, On a modified conjecture of De Giorgi, Math. Z., 254 (2006), 675-714. doi: 10.1007/s00209-006-0002-6.  Google Scholar

[44]

E. Sandier and S. Serfaty, Gamma-convergence of gradient flows with applications to Ginzburg-Landau, Comm. Pure Appl. Math, 57 (2004), 1627-1672. doi: 10.1002/cpa.20046.  Google Scholar

[45]

E. Sandier and S. Serfaty, Limiting vorticities for the Ginzburg-Landau equations, Duke Math J., 117 (2003), 403-446. doi: 10.1215/S0012-7094-03-11732-9.  Google Scholar

[46]

E. Sandier and S. Serfaty, A product-estimate for Ginzburg-Landau and corollaries, J. Func. Anal., 211 (2004), 219-244. doi: 10.1016/S0022-1236(03)00199-X.  Google Scholar

[47]

E. Sandier and S. Serfaty, "Vortices in the Magnetic Ginzburg-Landau Model," Progress in Nonlinear Differential Equations and their Applications, 70, Birkhäuser Boston, Inc., Boston, MA, 2007.  Google Scholar

[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), 561-592.  Google Scholar

[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), 1743-1751. doi: 10.1512/iumj.2008.57.3283.  Google Scholar

[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), 199-221. doi: 10.1512/iumj.2005.54.2497.  Google Scholar

[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), 177-217, Part II: The dynamics, Journal Eur. Math Society, 9 (2007), 383-426. doi: 10.4171/JEMS/84.  Google Scholar

[53]

C. Villani, "Optimal Transport. Old and New," Grundlehren der Mathematischen Wissenschaften, 338, Springer-Verlag, Berlin, 2009.  Google Scholar

[54]

M. G. Westdickenberg and Y. Tonegawa, Higher multiplicity in the one-dimensional Allen-Cahn action functional, Indiana Univ. Math. J., 56 (2007), 2935-2989. doi: 10.1512/iumj.2007.56.3182.  Google Scholar

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