Following Antman (1983, p. 283), the definition of a variational inequality is the following one.
Definition 1. Given a Banach space, a subset of , and a functional from to the dual space of the space ,
the variational inequality problem
is the problem of solving
for the variable belonging to the following inequality:
In general, the variational inequality problem can be formulated on any finite – or infinite-dimensionalBanach space. The three obvious steps in the study of the problem are the following ones:
Prove the existence of a solution: this step implies the mathematical correctness of the problem, showing that there is at least a solution.
Prove the uniqueness of the given solution: this step implies the physical correctness of the problem, showing that the solution can be used to represent a physical phenomenon. It is a particularly important step since most of the problems modeled by variational inequalities are of physical origin.
Find the solution or prove its regularity.
Examplesedit
The problem of finding the minimal value of a real-valued function of real variableedit
These necessary conditions can be summarized as the problem of finding such that
for
The absolute minimum must be searched between the solutions (if more than one) of the preceding inequality: note that the solution is a real number, therefore this is a finite dimensional variational inequality.
The general finite-dimensional variational inequalityedit
A formulation of the general problem in
is the following: given a subset of
and a mapping,
the finite-dimensional variational inequality problem associated with
consist of finding a -dimensionalvector belonging to
such that
Duvaut, Georges (1971), "Problèmes unilatéraux en mécanique des milieux continus", Actes du Congrès international des mathématiciens, 1970, ICM Proceedings, vol. Mathématiques appliquées (E), Histoire et Enseignement (F) – Volume 3, Paris: Gauthier-Villars, pp. 71–78, archived from the original (PDF) on 2015-07-25, retrieved 2015-07-25. A brief research survey describing the field of variational inequalities, precisely the sub-field of continuum mechanics problems with unilateral constraints.
Fichera, Gaetano (1995), "La nascita della teoria delle disequazioni variazionali ricordata dopo trent'anni", Incontro scientifico italo-spagnolo. Roma, 21 ottobre 1993, Atti dei Convegni Lincei (in Italian), vol. 114, Roma: Accademia Nazionale dei Lincei, pp. 47–53. The birth of the theory of variational inequalities remembered thirty years later (English translation of the title) is an historical paper describing the beginning of the theory of variational inequalities from the point of view of its founder.
Scientific worksedit
Facchinei, Francisco; Pang, Jong-Shi (2003), Finite Dimensional Variational Inequalities and Complementarity Problems, Vol. 1, Springer Series in Operations Research, Berlin–Heidelberg–New York: Springer-Verlag, ISBN 0-387-95580-1, Zbl 1062.90001
Facchinei, Francisco; Pang, Jong-Shi (2003), Finite Dimensional Variational Inequalities and Complementarity Problems, Vol. 2, Springer Series in Operations Research, Berlin–Heidelberg–New York: Springer-Verlag, ISBN 0-387-95581-X, Zbl 1062.90001
Fichera, Gaetano (1963), "Sul problema elastostatico di Signorini con ambigue condizioni al contorno" [On the elastostatic problem of Signorini with ambiguous boundary conditions], Rendiconti della Accademia Nazionale dei Lincei, Classe di Scienze Fisiche, Matematiche e Naturali, 8 (in Italian), 34 (2): 138–142, MR 0176661, Zbl 0128.18305. A short research note announcing and describing (without proofs) the solution of the Signorini problem.
Fichera, Gaetano (1964a), "Problemi elastostatici con vincoli unilaterali: il problema di Signorini con ambigue condizioni al contorno" [Elastostatic problems with unilateral constraints: the Signorini problem with ambiguous boundary conditions], Memorie della Accademia Nazionale dei Lincei, Classe di Scienze Fisiche, Matematiche e Naturali, 8 (in Italian), 7 (2): 91–140, Zbl 0146.21204. The first paper where an existence and uniqueness theorem for the Signorini problem is proved.
Fichera, Gaetano (1964b), "Elastostatic problems with unilateral constraints: the Signorini problem with ambiguous boundary conditions", Seminari dell'istituto Nazionale di Alta Matematica 1962–1963, Rome: Edizioni Cremonese, pp. 613–679. An English translation of (Fichera 1964a).
Roubíček, Tomáš (2013), Nonlinear Partial Differential Equations with Applications, ISNM. International Series of Numerical Mathematics, vol. 153 (2nd ed.), Basel–Boston–Berlin: Birkhäuser Verlag, pp. xx+476, doi:10.1007/978-3-0348-0513-1, ISBN 978-3-0348-0512-4, MR 3014456, Zbl 1270.35005.
Stampacchia, Guido (1964), "Formes bilineaires coercitives sur les ensembles convexes", Comptes rendus hebdomadaires des séances de l'Académie des sciences, 258: 4413–4416, Zbl 0124.06401, available at Gallica. The paper containing Stampacchia's generalization of the Lax–Milgram theorem.