# A converse to the Lions-Stampacchia theorem

ESAIM: Control, Optimisation and Calculus of Variations (2009)

- Volume: 15, Issue: 4, page 810-817
- ISSN: 1292-8119

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topErnst, Emil, and Théra, Michel. "A converse to the Lions-Stampacchia theorem." ESAIM: Control, Optimisation and Calculus of Variations 15.4 (2009): 810-817. <http://eudml.org/doc/245298>.

@article{Ernst2009,

abstract = {In this paper we show that a linear variational inequality over an infinite dimensional real Hilbert space admits solutions for every nonempty bounded closed and convex set, if and only if the linear operator involved in the variational inequality is pseudo-monotone in the sense of Brezis.},

author = {Ernst, Emil, Théra, Michel},

journal = {ESAIM: Control, Optimisation and Calculus of Variations},

keywords = {Lions-Stampacchia theorem; variational inequality; pseudo-monotone operator},

language = {eng},

number = {4},

pages = {810-817},

publisher = {EDP-Sciences},

title = {A converse to the Lions-Stampacchia theorem},

url = {http://eudml.org/doc/245298},

volume = {15},

year = {2009},

}

TY - JOUR

AU - Ernst, Emil

AU - Théra, Michel

TI - A converse to the Lions-Stampacchia theorem

JO - ESAIM: Control, Optimisation and Calculus of Variations

PY - 2009

PB - EDP-Sciences

VL - 15

IS - 4

SP - 810

EP - 817

AB - In this paper we show that a linear variational inequality over an infinite dimensional real Hilbert space admits solutions for every nonempty bounded closed and convex set, if and only if the linear operator involved in the variational inequality is pseudo-monotone in the sense of Brezis.

LA - eng

KW - Lions-Stampacchia theorem; variational inequality; pseudo-monotone operator

UR - http://eudml.org/doc/245298

ER -

## References

top- [1] H. Brezis, Équations et inéquations non linéaires dans les espaces vectoriels en dualité. Ann. Inst. Fourier 18 (1968) 115–175. Zbl0169.18602MR270222
- [2] G. Duvaut and J.L. Lions, Les inéquations en mécanique et en physique. Dunod, Paris (1972). Zbl0298.73001MR464857
- [3] G. Fichera, Problemi elastostatici con vincoli unilaterali: il problema die Signorini con ambigue condizioni al contorno. Mem. Accad. Naz. Lincei 8 (1964) 91–140. Zbl0146.21204MR178631
- [4] D. Goeleven and D. Motreanu, Variational and Hemivariational Inequalities: Theory, Methods, and Applications. Kluwer Academic Publishers (2003). Zbl1259.49002MR2036155
- [5] J.-L. Lions and G. Stampacchia, Variational inequalities. Comm. Pure Appl. Math. 20 (1967) 493–519. Zbl0152.34601MR216344
- [6] J.-L. Lions, E. Magenes, O.G. Mancino and S. Mazzone, Variational Analysis and Applications, in Proceedings of the 38th Conference of the School of Mathematics “G. Stampacchia”, in memory of Stampacchia and J.-L. Lions, Erice, June 20–July 1st 2003, F. Giannessi and A. Maugeri Eds., Nonconvex Optimization and its Applications 79, Springer-Verlag, New York (2005). Zbl1093.01535MR2160743
- [7] R.E. Showalter, Monotone Operators in Banach Space and Nonlinear Partial Differential Equations, Mathematical Surveys and Monographs 49. American Mathematical Society (1997). Zbl0870.35004MR1422252

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