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Singular limit of Allen--Cahn equation with constraint and its Lagrange multiplier
1. | Weierstrass Institute for Applied Analysis and Stochastics, Mohrenstrasse 39, 10117 Berlin, Germany |
2. | Department of Mathematics, Kyoto University of Education, Fuji 1, Fukakusa Fushimi-ku, Kyoto 612-8522 |
3. | Department of Mathematics, Faculty of Engineering, Kanagawa University, 3-27-1 Rokkakubashi, Kanagawa-ku, 221-8686 |
References:
[1] |
S. Allen and J. Cahn, A microscopic theory for antiphase boundary motion and its application to antiphase domain coarsening, Acta Metall., 27(1979), 1084-1095.
doi: 10.1016/0001-6160(79)90196-2. |
[2] |
L. Ambrosio, N. Fusco and D. Pallara, Functions of bounded variation and free discontinuity problems, Oxford Mathematical Monographs, The Clarendon Press, Oxford University Press, New York, 2000. |
[3] |
V. Barbu, Nonlinear differential equations of monotone types in Banach spaces, Springer Monographs in Mathematics, Springer, New York, 2010. |
[4] |
L. Blank, H. Garcke L. Sarbu and V. Styles, Primal-dual active set methods for Allen-Cahn variational inequalities with nonlocal constraints, Numer. Methods Partial Differential Equations, 29(2013), 999-1030. |
[5] |
V. Barbu and T. Precupanu, Convexity and optimization in Banach spaces, Fourth edition, Springer Monographs in Mathematics, Springer, Dordrecht, 2012. |
[6] |
H. Brézis, Opérateurs Maximaux Monotones et Semi-Groupes de Contractions dans les Espaces de Hilbert, North-Holland, Amsterdam, 1973. |
[7] |
H. Brézis, M. G. Crandall and A. Pazy, Perturbations of nonlinear maximal monotone sets in Banach space, Comm. Pure Appl. Math., 23(1970), 123-144. |
[8] |
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. |
[9] |
X. Chen and C. M. Elliott, Asymptotics for a parabolic double obstacle problem, Proc. Roy. Soc. London Ser. A, 444(1994), 429-445. |
[10] |
P. C. Fife, Dynamics of internal layers and diffusive interfaces, CBMS-NSF Regional Conf. Ser. in Appl. Math., 53, SIAM, Philadelphia, 1988. |
[11] |
T. Fukao and N. Kenmochi, Lagrange multipliers in variational inequalities for nonlinear operators of monotone type, Adv. Math. Sci. Appl., 23(2013), 545-574. |
[12] |
A. Ito, Asymptotic stability of Allen-Cahn model for nonlinear Laplacian with constraints, Adv. Math. Sci. Appl., 9(1999), 137-161. |
[13] |
A. Ito, N. Yamazaki and N. Kenmochi, Attractors of nonlinear evolution systems generated by time-dependent subdifferentials in Hilbert spaces, Dynamical systems and differential equations, Vol. I (Springfield, MO, 1996), Discrete Contin. Dynam. Systems 1998, Added Volume I, 327-349. |
[14] |
K. Ito and K. Kunisch, Lagrange multiplier approach to variational problems and applications, Advances in Design and Control, 15, Society for Industrial and Applied Mathematics (SIAM), Philadelphia, PA, 2008. |
[15] |
N. Kenmochi and K. Shirakawa, Stability for a parabolic variational inequality associated with total variation functional, Funkcial. Ekvac., 44(2001), 119-137. |
[16] |
T. Ohtsuka, K. Shirakawa and N. Yamazaki, Optimal control problem for Allen-Cahn type equation associated with total variation energy, Discrete Contin. Dyn. Syst. Ser. S, 5(2012), 159-181. |
[17] |
M. Ôtani, Nonmonotone perturbations for nonlinear, parabolic equations associated with subdifferential operators, Cauchy problems, J. Differential Equations, 46(1982), 268-299. |
[18] |
K. Shirakawa and M. Kimura, Stability analysis for Allen-Cahn type equation associated with the total variation energy, Nonlinear Anal., 60(2005), 257-282. |
[19] |
Y. Tonegawa, Integrality of varifolds in the singular limit of reaction-diffusion equations, Hiroshima Math. J., 33(2003), 323-341. |
show all references
References:
[1] |
S. Allen and J. Cahn, A microscopic theory for antiphase boundary motion and its application to antiphase domain coarsening, Acta Metall., 27(1979), 1084-1095.
doi: 10.1016/0001-6160(79)90196-2. |
[2] |
L. Ambrosio, N. Fusco and D. Pallara, Functions of bounded variation and free discontinuity problems, Oxford Mathematical Monographs, The Clarendon Press, Oxford University Press, New York, 2000. |
[3] |
V. Barbu, Nonlinear differential equations of monotone types in Banach spaces, Springer Monographs in Mathematics, Springer, New York, 2010. |
[4] |
L. Blank, H. Garcke L. Sarbu and V. Styles, Primal-dual active set methods for Allen-Cahn variational inequalities with nonlocal constraints, Numer. Methods Partial Differential Equations, 29(2013), 999-1030. |
[5] |
V. Barbu and T. Precupanu, Convexity and optimization in Banach spaces, Fourth edition, Springer Monographs in Mathematics, Springer, Dordrecht, 2012. |
[6] |
H. Brézis, Opérateurs Maximaux Monotones et Semi-Groupes de Contractions dans les Espaces de Hilbert, North-Holland, Amsterdam, 1973. |
[7] |
H. Brézis, M. G. Crandall and A. Pazy, Perturbations of nonlinear maximal monotone sets in Banach space, Comm. Pure Appl. Math., 23(1970), 123-144. |
[8] |
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. |
[9] |
X. Chen and C. M. Elliott, Asymptotics for a parabolic double obstacle problem, Proc. Roy. Soc. London Ser. A, 444(1994), 429-445. |
[10] |
P. C. Fife, Dynamics of internal layers and diffusive interfaces, CBMS-NSF Regional Conf. Ser. in Appl. Math., 53, SIAM, Philadelphia, 1988. |
[11] |
T. Fukao and N. Kenmochi, Lagrange multipliers in variational inequalities for nonlinear operators of monotone type, Adv. Math. Sci. Appl., 23(2013), 545-574. |
[12] |
A. Ito, Asymptotic stability of Allen-Cahn model for nonlinear Laplacian with constraints, Adv. Math. Sci. Appl., 9(1999), 137-161. |
[13] |
A. Ito, N. Yamazaki and N. Kenmochi, Attractors of nonlinear evolution systems generated by time-dependent subdifferentials in Hilbert spaces, Dynamical systems and differential equations, Vol. I (Springfield, MO, 1996), Discrete Contin. Dynam. Systems 1998, Added Volume I, 327-349. |
[14] |
K. Ito and K. Kunisch, Lagrange multiplier approach to variational problems and applications, Advances in Design and Control, 15, Society for Industrial and Applied Mathematics (SIAM), Philadelphia, PA, 2008. |
[15] |
N. Kenmochi and K. Shirakawa, Stability for a parabolic variational inequality associated with total variation functional, Funkcial. Ekvac., 44(2001), 119-137. |
[16] |
T. Ohtsuka, K. Shirakawa and N. Yamazaki, Optimal control problem for Allen-Cahn type equation associated with total variation energy, Discrete Contin. Dyn. Syst. Ser. S, 5(2012), 159-181. |
[17] |
M. Ôtani, Nonmonotone perturbations for nonlinear, parabolic equations associated with subdifferential operators, Cauchy problems, J. Differential Equations, 46(1982), 268-299. |
[18] |
K. Shirakawa and M. Kimura, Stability analysis for Allen-Cahn type equation associated with the total variation energy, Nonlinear Anal., 60(2005), 257-282. |
[19] |
Y. Tonegawa, Integrality of varifolds in the singular limit of reaction-diffusion equations, Hiroshima Math. J., 33(2003), 323-341. |
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