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, (1986).   Google Scholar

[2]

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

[3]

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

[4]

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

[6]

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

[8]

K. Ito and K. Kunisch, Lagrange multiplier approach to variational problems and applications,, Advances in Design and Control, (2008).   Google Scholar

[9]

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

[10]

N. Kenmochi, Monotonicity and compactness methods for nonlinear variational inequalities,, M. Chipot (Ed.), Vol.4 (2007), 203.   Google Scholar

[11]

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

[12]

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

[13]

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

[15]

Y. Yamada, On evolution equations generated by subdifferential operators,, J. Fac. Sci. Univ. Tokyo Sect. IA Math., 23 (1976), 491.   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, (1997), 288.   Google Scholar

show all references

References:
[1]

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

[2]

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

[3]

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

[4]

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

[6]

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

[8]

K. Ito and K. Kunisch, Lagrange multiplier approach to variational problems and applications,, Advances in Design and Control, (2008).   Google Scholar

[9]

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

[10]

N. Kenmochi, Monotonicity and compactness methods for nonlinear variational inequalities,, M. Chipot (Ed.), Vol.4 (2007), 203.   Google Scholar

[11]

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

[12]

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

[13]

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

[15]

Y. Yamada, On evolution equations generated by subdifferential operators,, J. Fac. Sci. Univ. Tokyo Sect. IA Math., 23 (1976), 491.   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, (1997), 288.   Google Scholar

[1]

Caiping Liu, Heungwing Lee. Lagrange multiplier rules for approximate solutions in vector optimization. Journal of Industrial & Management Optimization, 2012, 8 (3) : 749-764. doi: 10.3934/jimo.2012.8.749
[2]

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
[3]

T. A. Shaposhnikova, M. N. Zubova. Homogenization problem for a parabolic variational inequality with constraints on subsets situated on the boundary of the domain. Networks & Heterogeneous Media, 2008, 3 (3) : 675-689. doi: 10.3934/nhm.2008.3.675
[4]

Walter Allegretto, Yanping Lin, Shuqing Ma. On the box method for a non-local parabolic variational inequality. Discrete & Continuous Dynamical Systems - B, 2001, 1 (1) : 71-88. doi: 10.3934/dcdsb.2001.1.71
[5]

Pedro L. García, Antonio Fernández, César Rodrigo. Variational integrators for discrete Lagrange problems. Journal of Geometric Mechanics, 2010, 2 (4) : 343-374. doi: 10.3934/jgm.2010.2.343
[6]

Stefano Bianchini. On the Euler-Lagrange equation for a variational problem. Discrete & Continuous Dynamical Systems - A, 2007, 17 (3) : 449-480. doi: 10.3934/dcds.2007.17.449
[7]

Takeshi Fukao. Variational inequality for the Stokes equations with constraint. Conference Publications, 2011, 2011 (Special) : 437-446. doi: 10.3934/proc.2011.2011.437
[8]

Anthony Bloch, Leonardo Colombo, Fernando Jiménez. The variational discretization of the constrained higher-order Lagrange-Poincaré equations. Discrete & Continuous Dynamical Systems - A, 2019, 39 (1) : 309-344. doi: 10.3934/dcds.2019013
[9]

Benny Avelin, Tuomo Kuusi, Mikko Parviainen. Variational parabolic capacity. Discrete & Continuous Dynamical Systems - A, 2015, 35 (12) : 5665-5688. doi: 10.3934/dcds.2015.35.5665
[10]

S. J. Li, Z. M. Fang. On the stability of a dual weak vector variational inequality problem. Journal of Industrial & Management Optimization, 2008, 4 (1) : 155-165. doi: 10.3934/jimo.2008.4.155
[11]

Junkee Jeon, Jehan Oh. Valuation of American strangle option: Variational inequality approach. Discrete & Continuous Dynamical Systems - B, 2019, 24 (2) : 755-781. doi: 10.3934/dcdsb.2018206
[12]

Yutian Lei, Zhongxue Lü. Axisymmetry of locally bounded solutions to an Euler-Lagrange system of the weighted Hardy-Littlewood-Sobolev inequality. Discrete & Continuous Dynamical Systems - A, 2013, 33 (5) : 1987-2005. doi: 10.3934/dcds.2013.33.1987
[13]

Wenhui Shi. An epiperimetric inequality approach to the parabolic Signorini problem. Discrete & Continuous Dynamical Systems - A, 2020, 40 (3) : 1813-1846. doi: 10.3934/dcds.2020095
[14]

Masao Fukushima. A class of gap functions for quasi-variational inequality problems. Journal of Industrial & Management Optimization, 2007, 3 (2) : 165-171. doi: 10.3934/jimo.2007.3.165
[15]

Wenyan Zhang, Shu Xu, Shengji Li, Xuexiang Huang. Generalized weak sharp minima of variational inequality problems with functional constraints. Journal of Industrial & Management Optimization, 2013, 9 (3) : 621-630. doi: 10.3934/jimo.2013.9.621
[16]

Li Wang, Yang Li, Liwei Zhang. A differential equation method for solving box constrained variational inequality problems. Journal of Industrial & Management Optimization, 2011, 7 (1) : 183-198. doi: 10.3934/jimo.2011.7.183
[17]

Frank Jochmann. A variational inequality in Bean's model for superconductors with displacement current. Discrete & Continuous Dynamical Systems - A, 2009, 25 (2) : 545-565. doi: 10.3934/dcds.2009.25.545
[18]

Junfeng Yang. Dynamic power price problem: An inverse variational inequality approach. Journal of Industrial & Management Optimization, 2008, 4 (4) : 673-684. doi: 10.3934/jimo.2008.4.673
[19]

Hui-Qiang Ma, Nan-Jing Huang. Neural network smoothing approximation method for stochastic variational inequality problems. Journal of Industrial & Management Optimization, 2015, 11 (2) : 645-660. doi: 10.3934/jimo.2015.11.645
[20]

Ren-You Zhong, Nan-Jing Huang. Strict feasibility for generalized mixed variational inequality in reflexive Banach spaces. Numerical Algebra, Control & Optimization, 2011, 1 (2) : 261-274. doi: 10.3934/naco.2011.1.261

 Impact Factor: 

Tools

Metrics

  • PDF downloads (17)
  • HTML views (0)
  • Cited by (0)

Other articles
by authors

[Back to Top]

Copyright © 2019 American Institute of Mathematical Sciences