Previous |  Up |  Next


Lagrange multiplier; parabolic variational inequality
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.
[1] Baiocchi, C., Capelo, A.: Variational and Quasivariational Inequalities. Applications to Free Boundary Problems. Translated from the Italian. A Wiley-Interscience Publication Wiley, New York (1984). MR 0745619
[2] Barbu, V.: Nonlinear Differential Equations of Monotone Types in Banach Spaces. Springer Monographs in Mathematics Springer, Berlin (2010). MR 2582280 | Zbl 1197.35002
[3] Barbu, V., Precupanu, T.: Convexity and Optimization in Banach Spaces. 4th ed., Springer Monographs in Mathematics Springer, Dordrecht (2012). MR 3025420 | Zbl 1244.49001
[4] Brézis, H.: Problèmes unilatéraux. J. Math. pur. Appl., IX. Sér. 51 (1972), 1-168 French. MR 0428137 | Zbl 0237.35001
[5] Brézis, H.: Opérateurs Maximaux Monotones et Semi-Groupes de Contractions dans les Espaces de Hilbert. North-Holland Mathematics Studies 5 North-Holland, Amsterdam (1973), French. MR 0348562 | Zbl 0252.47055
[6] Fukao, T., Kenmochi, N.: Abstract theory of variational inequalities and Lagrange multipliers. Discrete Contin. Dyn. Syst., suppl. (2013), 237-246.
[7] Fukao, T., Kenmochi, N.: Lagrange multipliers in variational inequalities for nonlinear operators of monotone type. Adv. Math. Sci. Appl. 23 (2013), 545-574. MR 3236632 | Zbl 1300.47081
[8] Ito, K., Kunisch, K.: Lagrange Multiplier Approach to Variational Problems and Applications. Advances in Design and Control 15 SIAM, Philadelphia (2008). MR 2441683 | Zbl 1156.49002
[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).
[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. Zbl 0662.35054
[11] Kinderlehrer, D., Stampacchia, G.: An Introduction to Variational Inequalities and Their Applications. Pure and Applied Mathematics 88 Academic Press, New York (1980). MR 0567696 | Zbl 0457.35001
[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. MR 2919344 | Zbl 1230.65073
[13] Yamada, Y.: On evolution equations generated by subdifferential operators. J. Fac. Sci., Univ. Tokyo, Sect. I A 23 (1976), 491-515. MR 0425701 | Zbl 0343.34053
[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). MR 1658070
Partner of
EuDML logo