Abstract theory of variational inequalities and Lagrange multipliers

Takeshi Fukao; Nobuyuki Kenmochi

doi:10.3934/proc.2013.2013.237

Article Contents

2013, Volume 2013: 237-246. Doi: 10.3934/proc.2013.2013.237 open access

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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

Abstract / Introduction Related Papers Cited by

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.

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Mathematics Subject Classification: Primary: 47J20, 49J40; Secondary: 35K86.

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