American Institute of Mathematical Sciences

Advanced
2013, 2013(special): 237-246. doi: 10.3934/proc.2013.2013.237

Abstract theory of variational inequalities and Lagrange multipliers

Takeshi Fukao 1, and  Nobuyuki Kenmochi 2,

1. 

Department of Mathematics, Kyoto University of Education, Fuji 1, Fukakusa Fushimi-ku, Kyoto 612-8522
2. 

Department of Education, School of Education, Bukkyo University, 96 Kitahananobo-cho, Murasakino, Kita-ku, Kyoto, 603-8301

Received  October 2012 Revised  April 2013 Published  November 2013

In this paper, the existence and uniqueness questions of abstract parabolic variational inequalities are considered in connection with Lagrange multipliers. The focus of authors' attention is the characterization of parabolic variational inequalities by means of Lagrange multipliers. It is well-known that various kinds of parabolic differential equations under convex constraints are represented by variational inequalities with time-dependent constraints, and the usage of Lagrange multipliers associated with constraints makes it possible to reformulate the variational inequalities as equations. In this paper, as a typical case, a parabolic problem with nonlocal time-dependent obstacle is treated in the framework of abstract evolution equations governed by time-dependent subdifferentials.
Keywords: Lagrange multiplier., Parabolic variational inequality, subdifferential.
Mathematics Subject Classification: Primary: 47J20, 49J40; Secondary: 35K8.
Citation: Takeshi Fukao, Nobuyuki Kenmochi. Abstract theory of variational inequalities and Lagrange multipliers. Conference Publications, 2013, 2013 (special) : 237-246. doi: 10.3934/proc.2013.2013.237
References:
[1]

V. Barbu and Th. Precupanu, Convexity and optimization in Banach space, D. Reidel Publishing Company, Dordrecht, 1986.  Google Scholar

[2]

H. Brézis, Problèmes unilatéraux, J. Math. Pures Appl. (9), 51(1972), 1-168.  Google Scholar

[3]

H. Brézis, Un problème d'évolution avec contraintes unilatérales dépendant du temps, C. R. Acad. Sci. Paris, 274(1972), 310-313.  Google Scholar

[4]

H. Brézis, Opérateurs maximaux monotones et semi-groupes de contractions dans les espaces de Hilbert, North-Holland, Amsterdam, 1973.  Google Scholar

[5]

E. Ginder, Construction of solutions to heat-type problems with time-dependent volume constraints, Adv. Math. Sci. Appl., 20(2010), 467-482.  Google Scholar

[6]

E. Ginder and K. Švadlenka, The discrete Morse flow for volume-controlled membrane motions, Adv. Math. Sci. Appl., 22(2012), 1-19.  Google Scholar

[7]

A. Ito, N. Kenmochi and M. Niezgódka, Phase separation model of Penrose-Fife type with Signorini boundary condition, Adv. Math. Sci. Appl., 17(2007), 337-356.  Google Scholar

[8]

K. Ito and K. Kunisch, Lagrange multiplier approach to variational problems and applications, Advances in Design and Control, Society for Industrial and Applied Mathematics, Philadelphia, 2008.  Google Scholar

[9]

D. Kinderlehrer and G. Stampacchia, An introduction to variational inequalities and their applications, Academic Press, New York, (1980).  Google Scholar

[10]

N. Kenmochi, Monotonicity and compactness methods for nonlinear variational inequalities, M. Chipot (Ed.), Handbook of differential equations: Stationary partial differential equations, Vol.4, North-Holland, Amsterdam (2007), 203-298.  Google Scholar

[11]

N. Kenmochi, Solvability of nonlinear evolution equations with time-dependent constraints and applications, The Bull. Fac. Education, Chiba Univ., 30(1981), 1-86. Google Scholar

[12]

M. Kubo, The Cahn-Hilliard equation with time-dependent constraint, Nonlinear Anal., 75(2012), 5672-5685.  Google Scholar

[13]

M. D. P. Monteiro Marques, Differential inclusions in nonsmooth mechanical problems, shocks and dry friction, Progr. Nonlinear Differential Equations Appl., Birkhäuser, Boston, 1993.  Google Scholar

[14]

K. Švadlenka and S. Omata, Mathematical modelling of surface vibration with volume constraint and its analysis, Nonlinear Anal., 69(2008), 3202-3212.  Google Scholar

[15]

Y. Yamada, On evolution equations generated by subdifferential operators, J. Fac. Sci. Univ. Tokyo Sect. IA Math., 23(1976), 491-515.  Google Scholar

[16]

N. Yamazaki, A. Ito and N. Kenmochi, Global attractors of time-dependent double obstacle problems, pp. 288-301 in Functional analysis and global analysis, Springer, Singapore, 1997.  Google Scholar

show all references

References:
[1]

V. Barbu and Th. Precupanu, Convexity and optimization in Banach space, D. Reidel Publishing Company, Dordrecht, 1986.  Google Scholar

[2]

H. Brézis, Problèmes unilatéraux, J. Math. Pures Appl. (9), 51(1972), 1-168.  Google Scholar

[3]

H. Brézis, Un problème d'évolution avec contraintes unilatérales dépendant du temps, C. R. Acad. Sci. Paris, 274(1972), 310-313.  Google Scholar

[4]

H. Brézis, Opérateurs maximaux monotones et semi-groupes de contractions dans les espaces de Hilbert, North-Holland, Amsterdam, 1973.  Google Scholar

[5]

E. Ginder, Construction of solutions to heat-type problems with time-dependent volume constraints, Adv. Math. Sci. Appl., 20(2010), 467-482.  Google Scholar

[6]

E. Ginder and K. Švadlenka, The discrete Morse flow for volume-controlled membrane motions, Adv. Math. Sci. Appl., 22(2012), 1-19.  Google Scholar

[7]

A. Ito, N. Kenmochi and M. Niezgódka, Phase separation model of Penrose-Fife type with Signorini boundary condition, Adv. Math. Sci. Appl., 17(2007), 337-356.  Google Scholar

[8]

K. Ito and K. Kunisch, Lagrange multiplier approach to variational problems and applications, Advances in Design and Control, Society for Industrial and Applied Mathematics, Philadelphia, 2008.  Google Scholar

[9]

D. Kinderlehrer and G. Stampacchia, An introduction to variational inequalities and their applications, Academic Press, New York, (1980).  Google Scholar

[10]

N. Kenmochi, Monotonicity and compactness methods for nonlinear variational inequalities, M. Chipot (Ed.), Handbook of differential equations: Stationary partial differential equations, Vol.4, North-Holland, Amsterdam (2007), 203-298.  Google Scholar

[11]

N. Kenmochi, Solvability of nonlinear evolution equations with time-dependent constraints and applications, The Bull. Fac. Education, Chiba Univ., 30(1981), 1-86. Google Scholar

[12]

M. Kubo, The Cahn-Hilliard equation with time-dependent constraint, Nonlinear Anal., 75(2012), 5672-5685.  Google Scholar

[13]

M. D. P. Monteiro Marques, Differential inclusions in nonsmooth mechanical problems, shocks and dry friction, Progr. Nonlinear Differential Equations Appl., Birkhäuser, Boston, 1993.  Google Scholar

[14]

K. Švadlenka and S. Omata, Mathematical modelling of surface vibration with volume constraint and its analysis, Nonlinear Anal., 69(2008), 3202-3212.  Google Scholar

[15]

Y. Yamada, On evolution equations generated by subdifferential operators, J. Fac. Sci. Univ. Tokyo Sect. IA Math., 23(1976), 491-515.  Google Scholar

[16]

N. Yamazaki, A. Ito and N. Kenmochi, Global attractors of time-dependent double obstacle problems, pp. 288-301 in Functional analysis and global analysis, Springer, Singapore, 1997.  Google Scholar

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