# A converse to the Lions-Stampacchia Theorem

ESAIM: Control, Optimisation and Calculus of Variations (2008)

- 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 (2008): 810-817. <http://eudml.org/doc/90938>.

@article{Ernst2008,

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; Lions-Stampacchia theorem},

language = {eng},

month = {8},

number = {4},

pages = {810-817},

publisher = {EDP Sciences},

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

url = {http://eudml.org/doc/90938},

volume = {15},

year = {2008},

}

TY - JOUR

AU - Ernst, Emil

AU - Théra, Michel

TI - A converse to the Lions-Stampacchia Theorem

JO - ESAIM: Control, Optimisation and Calculus of Variations

DA - 2008/8//

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; Lions-Stampacchia theorem

UR - http://eudml.org/doc/90938

ER -

## References

top- H. Brezis, Équations et inéquations non linéaires dans les espaces vectoriels en dualité. Ann. Inst. Fourier18 (1968) 115–175. Zbl0169.18602
- G. Duvaut and J.L. Lions, Les inéquations en mécanique et en physique. Dunod, Paris (1972). Zbl0298.73001
- G. Fichera, Problemi elastostatici con vincoli unilaterali: il problema die Signorini con ambigue condizioni al contorno. Mem. Accad. Naz. Lincei8 (1964) 91–140. Zbl0146.21204
- D. Goeleven and D. Motreanu, Variational and Hemivariational Inequalities: Theory, Methods, and Applications. Kluwer Academic Publishers (2003). Zbl1259.49001
- J.-L. Lions and G. Stampacchia, Variational inequalities. Comm. Pure Appl. Math.20 (1967) 493–519. Zbl0152.34601
- 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 Applications79, Springer-Verlag, New York (2005).
- R.E. Showalter, Monotone Operators in Banach Space and Nonlinear Partial Differential Equations, Mathematical Surveys and Monographs49. American Mathematical Society (1997).

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