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

Abstract theory of variational inequalities and Lagrange multipliers

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.
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
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show all references

References:
[1]

D. Reidel Publishing Company, Dordrecht, 1986.  Google Scholar

[2]

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

[3]

C. R. Acad. Sci. Paris, 274(1972), 310-313.  Google Scholar

[4]

North-Holland, Amsterdam, 1973.  Google Scholar

[5]

Adv. Math. Sci. Appl., 20(2010), 467-482.  Google Scholar

[6]

Adv. Math. Sci. Appl., 22(2012), 1-19.  Google Scholar

[7]

Adv. Math. Sci. Appl., 17(2007), 337-356.  Google Scholar

[8]

Advances in Design and Control, Society for Industrial and Applied Mathematics, Philadelphia, 2008.  Google Scholar

[9]

Academic Press, New York, (1980).  Google Scholar

[10]

M. Chipot (Ed.), Handbook of differential equations: Stationary partial differential equations, Vol.4, North-Holland, Amsterdam (2007), 203-298.  Google Scholar

[11]

The Bull. Fac. Education, Chiba Univ., 30(1981), 1-86. Google Scholar

[12]

Nonlinear Anal., 75(2012), 5672-5685.  Google Scholar

[13]

Progr. Nonlinear Differential Equations Appl., Birkhäuser, Boston, 1993.  Google Scholar

[14]

Nonlinear Anal., 69(2008), 3202-3212.  Google Scholar

[15]

J. Fac. Sci. Univ. Tokyo Sect. IA Math., 23(1976), 491-515.  Google Scholar

[16]

pp. 288-301 in Functional analysis and global analysis, Springer, Singapore, 1997.  Google Scholar

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