Abstract theory of variational inequalities with Lagrange multipliers and application to nonlinear PDEs

Takeshi Fukao; Nobuyuki Kenmochi

Mathematica Bohemica (2014)

  • Volume: 139, Issue: 2, page 391-399
  • ISSN: 0862-7959

Abstract

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Recently, we established some generalizations of the theory of Lagrange multipliers arising from nonlinear programming in Banach spaces, which enable us to treat not only elliptic problems but also parabolic problems in the same generalized framework. The main objective of the present paper is to discuss a typical time-dependent double obstacle problem as a new application of the above mentioned generalization. Actually, we describe it as a usual parabolic variational inequality and then characterize it as a parabolic inclusion by using the Lagrange multiplier and the nonlinear maximal monotone operator associated with the time differential under time-dependent double obstacles.

How to cite

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Fukao, Takeshi, and Kenmochi, Nobuyuki. "Abstract theory of variational inequalities with Lagrange multipliers and application to nonlinear PDEs." Mathematica Bohemica 139.2 (2014): 391-399. <http://eudml.org/doc/261906>.

@article{Fukao2014,
abstract = {Recently, we established some generalizations of the theory of Lagrange multipliers arising from nonlinear programming in Banach spaces, which enable us to treat not only elliptic problems but also parabolic problems in the same generalized framework. The main objective of the present paper is to discuss a typical time-dependent double obstacle problem as a new application of the above mentioned generalization. Actually, we describe it as a usual parabolic variational inequality and then characterize it as a parabolic inclusion by using the Lagrange multiplier and the nonlinear maximal monotone operator associated with the time differential under time-dependent double obstacles.},
author = {Fukao, Takeshi, Kenmochi, Nobuyuki},
journal = {Mathematica Bohemica},
keywords = {Lagrange multiplier; parabolic variational inequality; Lagrange multiplier; parabolic variational inequality},
language = {eng},
number = {2},
pages = {391-399},
publisher = {Institute of Mathematics, Academy of Sciences of the Czech Republic},
title = {Abstract theory of variational inequalities with Lagrange multipliers and application to nonlinear PDEs},
url = {http://eudml.org/doc/261906},
volume = {139},
year = {2014},
}

TY - JOUR
AU - Fukao, Takeshi
AU - Kenmochi, Nobuyuki
TI - Abstract theory of variational inequalities with Lagrange multipliers and application to nonlinear PDEs
JO - Mathematica Bohemica
PY - 2014
PB - Institute of Mathematics, Academy of Sciences of the Czech Republic
VL - 139
IS - 2
SP - 391
EP - 399
AB - Recently, we established some generalizations of the theory of Lagrange multipliers arising from nonlinear programming in Banach spaces, which enable us to treat not only elliptic problems but also parabolic problems in the same generalized framework. The main objective of the present paper is to discuss a typical time-dependent double obstacle problem as a new application of the above mentioned generalization. Actually, we describe it as a usual parabolic variational inequality and then characterize it as a parabolic inclusion by using the Lagrange multiplier and the nonlinear maximal monotone operator associated with the time differential under time-dependent double obstacles.
LA - eng
KW - Lagrange multiplier; parabolic variational inequality; Lagrange multiplier; parabolic variational inequality
UR - http://eudml.org/doc/261906
ER -

References

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  8. Ito, K., Kunisch, K., Lagrange Multiplier Approach to Variational Problems and Applications, Advances in Design and Control 15 SIAM, Philadelphia (2008). (2008) Zbl1156.49002MR2441683
  9. Kenmochi, N., Résolution de Problèmes Variationnels Paraboliques non Linéaires par les Méthodes de Compacité et Monotonie, Thesis, Université Pierre et Marie Curie, Paris 6 French (1979). (1979) 
  10. Kenmochi, N., Solvability of nonlinear evolution equations with time-dependent constraints and applications, Bull. Fac. Educ., Chiba Univ., Part II 30 (1981), 1-87. (1981) Zbl0662.35054
  11. Kinderlehrer, D., Stampacchia, G., An Introduction to Variational Inequalities and Their Applications, Pure and Applied Mathematics 88 Academic Press, New York (1980). (1980) Zbl0457.35001MR0567696
  12. Kokurin, M. Yu., An exact penalty method for monotone variational inequalities and order optimal algorithms for finding saddle points, Russ. Math. 55 (2011), 19-27. (2011) Zbl1230.65073MR2919344
  13. Yamada, Y., On evolution equations generated by subdifferential operators, J. Fac. Sci., Univ. Tokyo, Sect. I A 23 (1976), 491-515. (1976) Zbl0343.34053MR0425701
  14. Yamazaki, N., Ito, A., Kenmochi, N., Global attractors of time-dependent double obstacle problems, Functional Analysis and Global Analysis T. Sunada et al. Springer, Singapore 288-301 (1997). (1997) MR1658070

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