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In mathematics, a variational inequality is an inequality involving a functional, which has to be solved for all possible values of a given variable, belonging usually to a convex set. The mathematical theory of variational inequalities was initially developed to deal with equilibrium problems, precisely the Signorini problem: in that model problem, the functional involved was obtained as the first variation of the involved potential energy. Therefore it has a variational origin, recalled by the name of the general abstract problem. The applicability of the theory has since been expanded to include problems from economics, finance, optimization and game theory.

History

The first problem involving a variational inequality was the Signorini problem, posed by Antonio Signorini in 1959 and solved by Gaetano Fichera in 1963, according to the references (Antman 1983, pp. 282–284) and (Fichera 1995): the first papers of the theory were (Fichera 1963) and (Fichera 1964a), (Fichera 1964b). Later on, Guido Stampacchia proved his generalization to the Lax–Milgram theorem in (Stampacchia 1964) in order to study the regularity problem for partial differential equations and coined the name "variational inequality" for all the problems involving inequalities of this kind. Georges Duvaut encouraged his graduate students to study and expand on Fichera's work, after attending a conference in Brixen on 1965 where Fichera presented his study of the Signorini problem, as Antman 1983, p. 283 reports: thus the theory become widely known throughout France. Also in 1965, Stampacchia and Jacques-Louis Lions extended earlier results of (Stampacchia 1964), announcing them in the paper (Lions & Stampacchia 1965): full proofs of their results appeared later in the paper (Lions & Stampacchia 1967).
Definition

Following Antman (1983, p. 283), the formal definition of a variational inequality is the following one.

Definition 1. Given a Banach space E$$\boldsymbol{E},$$ a subset $${\boldsymbol {K}}$$ of $$\boldsymbol{E}$$ , and a functional $${\displaystyle F\colon {\boldsymbol {K}}\to {\boldsymbol {E}}^{\ast }}$$ from $${\boldsymbol {K}}$$ to the dual space $${\displaystyle {\boldsymbol {E}}^{\ast }}$$ of the space $$\boldsymbol{E}$$ , the variational inequality problem is the problem of solving for the variable x belonging to $${\boldsymbol {K}}$$ the following inequality:

$$\langle F(x),y-x\rangle \geq 0\qquad \forall y\in {\boldsymbol {K}} where \( {\displaystyle \langle \cdot ,\cdot \rangle \colon {\boldsymbol {E}}^{\ast }\times {\boldsymbol {E}}\to \mathbb {R} }$$ is the duality pairing.

In general, the variational inequality problem can be formulated on any finite – or infinite-dimensional Banach 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.

Examples
The problem of finding the minimal value of a real-valued function of real variable

This is a standard example problem, reported by Antman (1983, p. 283): consider the problem of finding the minimal value of a differentiable function f over a closed interval I = [a,b]. Let $$x^{\ast }$$ be a point in I where the minimum occurs. Three cases can occur:

if $${\displaystyle a<x^{\ast }<b,}$$ then $${\displaystyle f^{\prime }(x^{\ast })=0;}$$
if $${\displaystyle x^{\ast }=a,}$$ then$${\displaystyle f^{\prime }(x^{\ast })\geq 0;}$$
if $${\displaystyle x^{\ast }=b,}$$ then $${\displaystyle f^{\prime }(x^{\ast })\leq 0.}$$

These necessary conditions can be summarized as the problem of finding $${\displaystyle x^{\ast }\in I}$$ such that

$${\displaystyle f^{\prime }(x^{\ast })(y-x^{\ast })\geq 0\quad }$$ for $${\displaystyle \quad \forall y\in I.}$$

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 inequality

A formulation of the general problem in $$\mathbb {R} ^{n}$$ is the following: given a subset K of $$\mathbb {R} ^{n}$$ and a mapping $${\displaystyle F\colon K\to \mathbb {R} ^{n}}$$, the finite-dimensional variational inequality problem associated with K consist of finding a n-dimensional vector x belonging to K such that

$$\langle F(x),y-x\rangle \geq 0\qquad \forall y\in K$$

where $${\displaystyle \langle \cdot ,\cdot \rangle \colon \mathbb {R} ^{n}\times \mathbb {R} ^{n}\to \mathbb {R} }$$ is the standard inner product on the vector space $$\mathbb {R} ^{n}.$$
The variational inequality for the Signorini problem
The classical Signorini problem: what will be the equilibrium configuration of the orange spherically shaped elastic body resting on the blue rigid frictionless plane?

In the historical survey (Fichera 1995), Gaetano Fichera describes the genesis of his solution to the Signorini problem: the problem consist in finding the elastic equilibrium configuration $${\displaystyle {\boldsymbol {u}}({\boldsymbol {x}})=\left(u_{1}({\boldsymbol {x}}),u_{2}({\boldsymbol {x}}),u_{3}({\boldsymbol {x}})\right)}$$ of an anisotropic non-homogeneous elastic body that lies in a subset A of the three-dimensional euclidean space whose boundary is $$\partial A$$ , resting on a rigid frictionless surface and subject only to its mass forces. The solution u of the problem exists and is unique (under precise assumptions) in the set of admissible displacements $${\displaystyle {\mathcal {U}}_{\Sigma }}$$ i.e. the set of displacement vectors satisfying the system of ambiguous boundary conditions if and only if

$$B({\boldsymbol {u}},{\boldsymbol {v}})-F({\boldsymbol {v}})\geq 0\qquad \forall {\boldsymbol {v}}\in {\mathcal {U}}_{\Sigma }$$

where $${\displaystyle B({\boldsymbol {u}},{\boldsymbol {v}})}$$ and $${\displaystyle F({\boldsymbol {v}})}$$ are the following functionals, written using the Einstein notation

$${\displaystyle B({\boldsymbol {u}},{\boldsymbol {v}})=-\int _{A}\sigma _{ik}({\boldsymbol {u}})\varepsilon _{ik}({\boldsymbol {v}})\,\mathrm {d} x},$$ $${\displaystyle F({\boldsymbol {v}})=\int _{A}v_{i}f_{i}\,\mathrm {d} x+\int _{\partial A\setminus \Sigma }\!\!\!\!\!v_{i}g_{i}\,\mathrm {d} \sigma },$$ $${\displaystyle {\boldsymbol {u}},{\boldsymbol {v}}\in {\mathcal {U}}_{\Sigma }}$$

where, for all $${\displaystyle {\boldsymbol {x}}\in A},$$

$$\Sigma$$ is the contact surface (or more generally a contact set),
$${\displaystyle {\boldsymbol {f}}({\boldsymbol {x}})=\left(f_{1}({\boldsymbol {x}}),f_{2}({\boldsymbol {x}}),f_{3}({\boldsymbol {x}})\right)}$$ is the body force applied to the body,
$${\displaystyle {\boldsymbol {g}}({\boldsymbol {x}})=\left(g_{1}({\boldsymbol {x}}),g_{2}({\boldsymbol {x}}),g_{3}({\boldsymbol {x}})\right)}$$ is the surface force applied to $${\displaystyle \partial A\!\setminus \!\Sigma },$$
$${\displaystyle {\boldsymbol {\varepsilon }}={\boldsymbol {\varepsilon }}({\boldsymbol {u}})=\left(\varepsilon _{ik}({\boldsymbol {u}})\right)=\left({\frac {1}{2}}\left({\frac {\partial u_{i}}{\partial x_{k}}}+{\frac {\partial u_{k}}{\partial x_{i}}}\right)\right)}$$ is the infinitesimal strain tensor,
$${\displaystyle {\boldsymbol {\sigma }}=\left(\sigma _{ik}\right)}$$ is the Cauchy stress tensor, defined as

$$\sigma _{{ik}}=-{\frac {\partial W}{\partial \varepsilon _{{ik}}}}\qquad \forall i,k=1,2,3$$

where $$W(\boldsymbol{\varepsilon})=a_{ikjh}(\boldsymbol{x})\varepsilon_{ik}\varepsilon_{ik}$$ is the elastic potential energy and $${\displaystyle {\boldsymbol {a}}({\boldsymbol {x}})=\left(a_{ikjh}({\boldsymbol {x}})\right)}$$ is the elasticity tensor.

Complementarity theory
Differential variational inequality
Extended Mathematical Programming for Equilibrium Problems
Mathematical programming with equilibrium constraints
Obstacle problem
Projected dynamical system
Signorini problem
Unilateral contact

References
Historical references

Antman, Stuart (1983), "The influence of elasticity in analysis: modern developments", Bulletin of the American Mathematical Society, 9 (3): 267–291, doi:10.1090/S0273-0979-1983-15185-6, MR 0714990, Zbl 0533.73001. An historical paper about the fruitful interaction of elasticity theory and mathematical analysis: the creation of the theory of variational inequalities by Gaetano Fichera is described in §5, pages 282–284.
Duvaut, Georges (1971), "Problèmes unilatéraux en mécanique des milieux continus", Actes du Congrès international des mathématiciens, 1970, ICM Proceedings, 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), 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 works

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", Rendiconti della Accademia Nazionale dei Lincei, Classe di Scienze Fisiche, Matematiche e Naturali, 8 (in Italian), 34 (2): 138–142, Zbl 0128.18305. "On the elastostatic problem of Signorini with ambiguous boundary conditions" (English translation of the title) is a short research note announcing and describing the solution of the Signorini problem.
Fichera, Gaetano (1964a), "Problemi elastostatici con vincoli unilaterali: il problema di Signorini con ambigue condizioni al contorno", Memorie della Accademia Nazionale dei Lincei, Classe di Scienze Fisiche, Matematiche e Naturali, 8 (in Italian), 7 (2): 91–140, Zbl 0146.21204. "Elastostatic problems with unilateral constraints: the Signorini problem with ambiguous boundary conditions" (English translation of the title) is 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).
Glowinski, Roland; Lions, Jacques-Louis; Trémolières, Raymond (1981), Numerical analysis of variational inequalities. Translated from the French, Studies in Mathematics and its Applications, 8, Amsterdam–New York–Oxford: North-Holland, pp. xxix+776, ISBN 0-444-86199-8, MR 0635927, Zbl 0463.65046
Kinderlehrer, David; Stampacchia, Guido (1980), An Introduction to Variational Inequalities and Their Applications, Pure and Applied Mathematics, 88, Boston–London–New York–San Diego–Sydney–Tokyo–Toronto: Academic Press, ISBN 0-89871-466-4, Zbl 0457.35001.
Lions, Jacques-Louis; Stampacchia, Guido (1965), "Inéquations variationnelles non coercives", Comptes rendus hebdomadaires des séances de l'Académie des sciences, 261: 25–27, Zbl 0136.11906, available at Gallica. Announcements of the results of paper (Lions & Stampacchia 1967).
Lions, Jacques-Louis; Stampacchia, Guido (1967), "Variational inequalities", Communications on Pure and Applied Mathematics, 20 (3): 493–519, doi:10.1002/cpa.3160200302, Zbl 0152.34601, archived from the original on 2013-01-05 External link in |journal= (help). An important paper, describing the abstract approach of the authors to the theory of variational inequalities.
Roubíček, Tomáš (2013), Nonlinear Partial Differential Equations with Applications, ISNM. International Series of Numerical Mathematics, 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.

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