Abstract theory of variational inequalities and Lagrange multipliers

Pages: 237 - 246, Issue special, November 2013

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Takeshi Fukao - Department of Mathematics, Kyoto University of Education, Fuji 1, Fukakusa Fushimi-ku, Kyoto 612-8522, Japan (email)
Nobuyuki Kenmochi - Department of Education, School of Education, Bukkyo University, 96 Kitahananobo-cho, Murasakino, Kita-ku, Kyoto, 603-8301, Japan (email)

Abstract: 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:  Parabolic variational inequality, subdifferential, Lagrange multiplier.
Mathematics Subject Classification:  Primary: 47J20, 49J40; Secondary: 35K86.

Received: October 2012;      Revised: April 2013;      Published: November 2013.