# American Institute of Mathematical Sciences

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

## 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

Received  September 2014 Revised  January 2015 Published  November 2015

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.
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.

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. 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.
 [1] Cristina Pocci. On singular limit of a nonlinear $p$-order equation related to Cahn-Hilliard and Allen-Cahn evolutions. Evolution Equations & Control Theory, 2013, 2 (3) : 517-530. doi: 10.3934/eect.2013.2.517 [2] Alain Miranville, Wafa Saoud, Raafat Talhouk. On the Cahn-Hilliard/Allen-Cahn equations with singular potentials. Discrete & Continuous Dynamical Systems - B, 2017, 22 (11) : 1-19. doi: 10.3934/dcdsb.2018308 [3] Gianni Gilardi. On an Allen-Cahn type integrodifferential equation. Discrete & Continuous Dynamical Systems - S, 2013, 6 (3) : 703-709. doi: 10.3934/dcdss.2013.6.703 [4] Georgia Karali, Yuko Nagase. On the existence of solution for a Cahn-Hilliard/Allen-Cahn equation. Discrete & Continuous Dynamical Systems - S, 2014, 7 (1) : 127-137. doi: 10.3934/dcdss.2014.7.127 [5] Hongmei Cheng, Rong Yuan. Multidimensional stability of disturbed pyramidal traveling fronts in the Allen-Cahn equation. Discrete & Continuous Dynamical Systems - B, 2015, 20 (4) : 1015-1029. doi: 10.3934/dcdsb.2015.20.1015 [6] Xinlong Feng, Huailing Song, Tao Tang, Jiang Yang. Nonlinear stability of the implicit-explicit methods for the Allen-Cahn equation. Inverse Problems & Imaging, 2013, 7 (3) : 679-695. doi: 10.3934/ipi.2013.7.679 [7] Christos Sourdis. On the growth of the energy of entire solutions to the vector Allen-Cahn equation. Communications on Pure & Applied Analysis, 2015, 14 (2) : 577-584. doi: 10.3934/cpaa.2015.14.577 [8] Paul H. Rabinowitz, Ed Stredulinsky. On a class of infinite transition solutions for an Allen-Cahn model equation. Discrete & Continuous Dynamical Systems - A, 2008, 21 (1) : 319-332. doi: 10.3934/dcds.2008.21.319 [9] Ciprian G. Gal, Maurizio Grasselli. The non-isothermal Allen-Cahn equation with dynamic boundary conditions. Discrete & Continuous Dynamical Systems - A, 2008, 22 (4) : 1009-1040. doi: 10.3934/dcds.2008.22.1009 [10] Eleonora Cinti. Saddle-shaped solutions for the fractional Allen-Cahn equation. Discrete & Continuous Dynamical Systems - S, 2018, 11 (3) : 441-463. doi: 10.3934/dcdss.2018024 [11] Zhuoran Du, Baishun Lai. Transition layers for an inhomogeneous Allen-Cahn equation in Riemannian manifolds. Discrete & Continuous Dynamical Systems - A, 2013, 33 (4) : 1407-1429. doi: 10.3934/dcds.2013.33.1407 [12] Charles-Edouard Bréhier, Ludovic Goudenège. Analysis of some splitting schemes for the stochastic Allen-Cahn equation. Discrete & Continuous Dynamical Systems - B, 2017, 22 (11) : 1-22. doi: 10.3934/dcdsb.2019077 [13] Changchun Liu, Hui Tang. Existence of periodic solution for a Cahn-Hilliard/Allen-Cahn equation in two space dimensions. Evolution Equations & Control Theory, 2017, 6 (2) : 219-237. doi: 10.3934/eect.2017012 [14] Fang Li, Kimie Nakashima. Transition layers for a spatially inhomogeneous Allen-Cahn equation in multi-dimensional domains. Discrete & Continuous Dynamical Systems - A, 2012, 32 (4) : 1391-1420. doi: 10.3934/dcds.2012.32.1391 [15] Takeshi Ohtsuka, Ken Shirakawa, Noriaki Yamazaki. Optimal control problem for Allen-Cahn type equation associated with total variation energy. Discrete & Continuous Dynamical Systems - S, 2012, 5 (1) : 159-181. doi: 10.3934/dcdss.2012.5.159 [16] Isabeau Birindelli, Enrico Valdinoci. On the Allen-Cahn equation in the Grushin plane: A monotone entire solution that is not one-dimensional. Discrete & Continuous Dynamical Systems - A, 2011, 29 (3) : 823-838. doi: 10.3934/dcds.2011.29.823 [17] Xufeng Xiao, Xinlong Feng, Jinyun Yuan. The stabilized semi-implicit finite element method for the surface Allen-Cahn equation. Discrete & Continuous Dynamical Systems - B, 2017, 22 (7) : 2857-2877. doi: 10.3934/dcdsb.2017154 [18] Giorgio Fusco. Layered solutions to the vector Allen-Cahn equation in $\mathbb{R}^2$. Minimizers and heteroclinic connections. Communications on Pure & Applied Analysis, 2017, 16 (5) : 1807-1841. doi: 10.3934/cpaa.2017088 [19] Xiaofeng Yang. Error analysis of stabilized semi-implicit method of Allen-Cahn equation. Discrete & Continuous Dynamical Systems - B, 2009, 11 (4) : 1057-1070. doi: 10.3934/dcdsb.2009.11.1057 [20] Giorgio Fusco, Francesco Leonetti, Cristina Pignotti. On the asymptotic behavior of symmetric solutions of the Allen-Cahn equation in unbounded domains in $\mathbb{R}^2$. Discrete & Continuous Dynamical Systems - A, 2017, 37 (2) : 725-742. doi: 10.3934/dcds.2017030

Impact Factor: