Singular limit of Allen--Cahn equation with constraint and its Lagrange multiplier

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2015, 2015(special): 418-427. doi: 10.3934/proc.2015.0418

Singular limit of Allen--Cahn equation with constraint and its Lagrange multiplier

Mohammad Hassan Farshbaf-Shaker ^1, , Takeshi Fukao ^2, and Noriaki Yamazaki ^3,

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

Received September 2014 Revised January 2015 Published November 2015

Abstract
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We consider the Allen--Cahn equation with a constraint. Our constraint is provided by the subdifferential of the indicator function on a closed interval, which is the multivalued function. In this paper we give the characterization of the Lagrange multiplier for our equation. Moreover, we consider the singular limit of our system and clarify the limit of the solution and the Lagrange multiplier for our problem.

Keywords: double obstacle, singular limit, constraint, Allen-Cahn equation, Lagrange multiplier, subdierential..

Mathematics Subject Classification: Primary: 35K57, 35R35; Secondary: 35B2.

Citation: Mohammad Hassan Farshbaf-Shaker, Takeshi Fukao, Noriaki Yamazaki. Singular limit of Allen--Cahn equation with constraint and its Lagrange multiplier. Conference Publications, 2015, 2015 (special) : 418-427. doi: 10.3934/proc.2015.0418

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. 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, (2000).
[3]	V. Barbu, Nonlinear differential equations of monotone types in Banach spaces,, Springer Monographs in Mathematics, (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.
[5]	V. Barbu and T. Precupanu, Convexity and optimization in Banach spaces, Fourth edition,, Springer Monographs in Mathematics, (2012).
[6]	H. Brézis, Opérateurs Maximaux Monotones et Semi-Groupes de Contractions dans les Espaces de Hilbert,, North-Holland, (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.
[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.
[9]	X. Chen and C. M. Elliott, Asymptotics for a parabolic double obstacle problem,, Proc. Roy. Soc. London Ser. A, 444 (1994), 429.
[10]	P. C. Fife, Dynamics of internal layers and diffusive interfaces,, CBMS-NSF Regional Conf. Ser. in Appl. Math., 53 (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.
[12]	A. Ito, Asymptotic stability of Allen-Cahn model for nonlinear Laplacian with constraints,, Adv. Math. Sci. Appl., 9 (1999), 137.
[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, (1996), 327.
[14]	K. Ito and K. Kunisch, Lagrange multiplier approach to variational problems and applications,, Advances in Design and Control, 15 (2008).
[15]	N. Kenmochi and K. Shirakawa, Stability for a parabolic variational inequality associated with total variation functional,, Funkcial. Ekvac., 44 (2001), 119.
[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.
[17]	M. Ôtani, Nonmonotone perturbations for nonlinear, parabolic equations associated with subdifferential operators,, Cauchy problems, 46 (1982), 268.
[18]	K. Shirakawa and M. Kimura, Stability analysis for Allen-Cahn type equation associated with the total variation energy,, Nonlinear Anal., 60 (2005), 257.
[19]	Y. Tonegawa, Integrality of varifolds in the singular limit of reaction-diffusion equations,, Hiroshima Math. J., 33 (2003), 323.

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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. 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, (2000).
[3]	V. Barbu, Nonlinear differential equations of monotone types in Banach spaces,, Springer Monographs in Mathematics, (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.
[5]	V. Barbu and T. Precupanu, Convexity and optimization in Banach spaces, Fourth edition,, Springer Monographs in Mathematics, (2012).
[6]	H. Brézis, Opérateurs Maximaux Monotones et Semi-Groupes de Contractions dans les Espaces de Hilbert,, North-Holland, (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.
[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.
[9]	X. Chen and C. M. Elliott, Asymptotics for a parabolic double obstacle problem,, Proc. Roy. Soc. London Ser. A, 444 (1994), 429.
[10]	P. C. Fife, Dynamics of internal layers and diffusive interfaces,, CBMS-NSF Regional Conf. Ser. in Appl. Math., 53 (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.
[12]	A. Ito, Asymptotic stability of Allen-Cahn model for nonlinear Laplacian with constraints,, Adv. Math. Sci. Appl., 9 (1999), 137.
[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, (1996), 327.
[14]	K. Ito and K. Kunisch, Lagrange multiplier approach to variational problems and applications,, Advances in Design and Control, 15 (2008).
[15]	N. Kenmochi and K. Shirakawa, Stability for a parabolic variational inequality associated with total variation functional,, Funkcial. Ekvac., 44 (2001), 119.
[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.
[17]	M. Ôtani, Nonmonotone perturbations for nonlinear, parabolic equations associated with subdifferential operators,, Cauchy problems, 46 (1982), 268.
[18]	K. Shirakawa and M. Kimura, Stability analysis for Allen-Cahn type equation associated with the total variation energy,, Nonlinear Anal., 60 (2005), 257.
[19]	Y. Tonegawa, Integrality of varifolds in the singular limit of reaction-diffusion equations,, Hiroshima Math. J., 33 (2003), 323.

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