View source for Variational inequality

In [[Mathematics|mathematics]], a '''variational inequality''' is an [[Inequality (mathematics)|inequality]] involving a [[Functional (mathematics)|functional]], which has to be [[Inequality (mathematics)#Solving Inequalities|solved]] for all possible values of a given [[Variable (mathematics)|variable]], belonging usually to a [[Convex set|convex set]]. The mathematical [[Theory|theory]] of variational inequalities was initially developed to deal with [[Equilibrium point|equilibrium]] problems, precisely the [[Physics:Signorini problem|Signorini problem]]: in that model problem, the functional involved was obtained as the [[First variation|first variation]] of the involved [[Physics:Signorini problem#The potential energy|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 [[Finance:Economics|economics]], [[Finance:Finance|finance]], optimization and [[Game theory|game theory]].
== History ==
The first problem involving a variational inequality was the [[Physics:Signorini problem|Signorini problem]], posed by [[Biography:Antonio Signorini (physicist)|Antonio Signorini]] in 1959 and solved by [[Biography:Gaetano Fichera|Gaetano Fichera]] in 1963, according to the references {{Harv|Antman|1983|pp=282–284}} and {{Harv|Fichera|1995}}: the first papers of the theory were {{Harv|Fichera|1963}} and {{Harv|Fichera|1964a}}, {{Harv|Fichera|1964b}}. Later on, [[Biography:Guido Stampacchia|Guido Stampacchia]] proved his generalization to the [[Lax–Milgram theorem]] in {{Harv|Stampacchia|1964}} in order to study the regularity problem for [[Partial differential equation|partial differential equation]]s and [[Finance:Coin|coin]]ed the name "variational inequality" for all the problems involving [[Inequality (mathematics)|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 {{Harvnb|Antman|1983|p=283}} reports: thus the theory become widely known throughout {{wipe|France}}. Also in 1965, Stampacchia and [[Biography:Jacques-Louis Lions|Jacques-Louis Lions]] extended earlier results of {{Harv|Stampacchia|1964}}, announcing them in the paper {{Harv|Lions|Stampacchia|1965}}: full proofs of their results appeared later in the paper {{Harv|Lions|Stampacchia|1967}}.

== Definition ==
Following {{Harvtxt|Antman|1983|p=283}}, the definition of a variational inequality is the following one.

{{EquationRef|1|Definition 1.}} Given a [[Banach space]] <math>\boldsymbol{E}</math>, a [[Subset|subset]] <math>\boldsymbol{K}</math> of <math>\boldsymbol{E}</math>, and a functional <math>F\colon \boldsymbol{K}\to \boldsymbol{E}^{\ast}</math> from <math>\boldsymbol{K}</math> to the [[Dual space|dual space]] <math>\boldsymbol{E}^{\ast}</math> of the space <math>\boldsymbol{E}</math>, 
the variational inequality problem 
is the problem of [[Inequality (mathematics)#Solving Inequalities|solving]] 
for the [[Variable (mathematics)|variable]] <math>x</math> belonging to <math>\boldsymbol{K}</math> the following [[Inequality (mathematics)|inequality]]:

:<math>\langle F(x), y-x \rangle \geq 0\qquad\forall y \in \boldsymbol{K}</math>

where <math>\langle\cdot,\cdot\rangle\colon 
\boldsymbol{E}^{\ast}\times\boldsymbol{E}\to
\mathbb{R}</math> 
is the [[Dual space|duality pairing]].

In general, the variational inequality problem can be formulated on any [[Finite set|finite]] – or [[Infinite set|infinite]]-[[Physics:Dimension|dimension]]al [[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 or prove its regularity.

==Examples==
===The problem of finding the minimal value of a real-valued function of real variable===
This is a standard example problem, reported by {{Harvtxt|Antman|1983|p=283}}: consider the problem of finding the minimal value of a  [[Differentiable function|differentiable function]] <math>f</math> over a closed interval <math>I = [a,b]</math>. Let <math>x^{\ast}</math> be a point in <math>I</math> where the minimum occurs. Three cases can occur:

# if <math>a<x^{\ast}< b,</math> then <math>f^{\prime}(x^{\ast}) = 0;</math>
# if <math>x^{\ast}=a,</math> then <math>f^{\prime}(x^{\ast}) \ge 0;</math>
# if <math>x^{\ast}=b,</math> then <math>f^{\prime}(x^{\ast}) \le 0.</math>

These necessary conditions can be summarized as the problem of finding  <math>x^{\ast}\in I</math> such that

:<math>f^{\prime}(x^{\ast})(y-x^{\ast}) \geq 0\quad</math> for <math>\quad\forall y \in I.</math>

The absolute minimum must be searched between the solutions (if more than one) of the preceding [[Inequality (mathematics)|inequality]]: note that the solution is a [[Real number|real number]], therefore this is a finite dimensional variational inequality.

===The general finite-dimensional variational inequality===
A formulation of the general problem in <math>\mathbb{R}^n</math> 
is the following: given a [[Subset|subset]] 
<math>K</math> of 
<math>\mathbb{R}^{n}</math> 
and a [[Map (mathematics)|mapping]] 
<math>F\colon K\to\mathbb{R}^{n}</math>, 
the [[Finite set|finite]]-[[Physics:Dimension|dimension]]al variational inequality problem associated with <math>K</math> 
consist of finding a [[Physics:Dimension|<math>n</math>-dimensional]] [[Euclidean vector|vector]] <math>x</math> belonging to 
<math>K</math> such that

:<math>\langle F(x), y-x \rangle \geq 0\qquad\forall y \in K</math>

where <math>\langle\cdot,\cdot\rangle\colon\mathbb{R}^{n}\times\mathbb{R}^{n}\to\mathbb{R}</math> 
is the standard inner product on the [[Vector space|vector space]] <math>\mathbb{R}^{n}</math>.

=== The variational inequality for the Signorini problem ===
[[File:Classical Signorini problem.svg|thumb|right|400px|The classical Signorini problem: what will be the equilibrium configuration of the orange spherically shaped elastic body resting on the blue [[Rigid body|rigid]] frictionless [[Plane (geometry)|plane]]?]]
In the historical survey {{Harv|Fichera|1995}}, [[Biography:Gaetano Fichera|Gaetano Fichera]] describes the genesis of his solution to the [[Physics:Signorini problem|Signorini problem]]: the problem consist in finding the [[Physics:Linear elasticity#Elastostatics|elastic equilibrium]] [[Physics:Continuum mechanics#Formulation of models|configuration]] 
<math>\boldsymbol{u}(\boldsymbol{x})
=\left(u_1(\boldsymbol{x}),u_2(\boldsymbol{x}),u_3(\boldsymbol{x})\right)</math> 
of an [[Anisotropy#Material science and engineering|anisotropic]] non-homogeneous [[Philosophy:Physical body|elastic body]] that lies in a [[Subset|subset]] 
<math>A</math> of the three-[[Physics:Dimension|dimension]]al [[Euclidean space|euclidean space]] whose [[Boundary (topology)|boundary]] is <math>\partial A</math>, resting on a [[Rigid body|rigid]] [[Physics:Frictionless plane|frictionless]] [[Surface (topology)|surface]] and subject only to its [[Physics:Weight|mass force]]s. 
The solution '''<math>u</math>''' of the problem exists and is unique (under precise assumptions) in the [[Set (mathematics)|set]] of '''admissible displacements''' 
<math>\mathcal{U}_\Sigma</math> 
i.e. the set of displacement vectors satisfying the system of [[Physics:Signorini problem#The ambiguous boundary conditions|ambiguous boundary conditions]] if and only if

:<math>B(\boldsymbol{u},\boldsymbol{v} - \boldsymbol{u}) - F(\boldsymbol{v} - \boldsymbol{u}) \geq 0 \qquad \forall \boldsymbol{v} \in \mathcal{U}_\Sigma </math>

where <math>B(\boldsymbol{u},\boldsymbol{v}) </math> 
and <math>F(\boldsymbol{v}) </math> 
are the following [[Functional (mathematics)|functionals]], 
written using the [[Einstein notation]]

:<math>B(\boldsymbol{u},\boldsymbol{v}) = -\int_A \sigma_{ik}(\boldsymbol{u})\varepsilon_{ik}(\boldsymbol{v})\,\mathrm{d}x</math>,&nbsp;&nbsp;&nbsp;  <math>F(\boldsymbol{v}) = \int_A v_i f_i\,\mathrm{d}x + \int_{\partial A\setminus\Sigma}\!\!\!\!\! v_i g_i \,\mathrm{d}\sigma</math>,&nbsp;&nbsp;&nbsp;  <math>\boldsymbol{u},\boldsymbol{v} \in \mathcal{U}_\Sigma </math>

where, for all <math>\boldsymbol{x}\in A</math>, 
*<math>\Sigma</math> is the contact [[Surface (topology)|surface]] (or more generally a contact [[Set (mathematics)|set]]),
*<math>\boldsymbol{f}(\boldsymbol{x}) = \left( f_1(\boldsymbol{x}), f_2(\boldsymbol{x}), f_3(\boldsymbol{x}) \right)</math> is the ''[[Physics:Body force|body force]]'' applied to the body,
*<math>\boldsymbol{g}(\boldsymbol{x})=\left(g_1(\boldsymbol{x}),g_2(\boldsymbol{x}),g_3(\boldsymbol{x})\right)</math> is the [[Physics:Surface force|surface force]] applied to <math>\partial A\!\setminus\!\Sigma</math>,
*<math>\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)</math> is the infinitesimal strain tensor,
*<math>\boldsymbol{\sigma}=\left(\sigma_{ik}\right)</math> is the [[Physics:Cauchy stress tensor|Cauchy stress tensor]], defined as
::<math>\sigma_{ik}= - \frac{\partial W}{\partial \varepsilon_{ik}} \qquad\forall i,k=1,2,3</math>

:where <math>W(\boldsymbol{\varepsilon})=a_{ikjh}(\boldsymbol{x})\varepsilon_{ik}\varepsilon_{jh}</math> <!---- may be more correct: \varepsilon_{ik}\varepsilon_{jh} ? ----> is the elastic potential energy and <math>\boldsymbol{a}(\boldsymbol{x})=\left(a_{ikjh}(\boldsymbol{x})\right)</math> is the [[Physics:Elasticity tensor|elasticity tensor]].

==See also==
*[[Complementarity theory]]
*[[Differential variational inequality]]
*[[Extended Mathematical Programming (EMP)#Equilibrium Problems|Extended Mathematical Programming for Equilibrium Problems]]
*[[Mathematical programming with equilibrium constraints]]
*[[Obstacle problem]]
*[[Projected dynamical system]]
*[[Physics:Signorini problem|Signorini problem]]
*[[Physics:Unilateral contact|Unilateral contact]]

== References ==
===Historical references===
*{{Citation
| last = Antman
| first = Stuart 
| title = The influence of elasticity in analysis: modern developments
| journal = [[Organization:Bulletin of the American Mathematical Society|Bulletin of the American Mathematical Society]]
| volume = 9
| issue = 3
| pages = 267–291
| year = 1983
| doi = 10.1090/S0273-0979-1983-15185-6 
| mr = 714990 
| zbl = 0533.73001
| doi-access = free
}}. An historical paper about the fruitful interaction of elasticity theory and [[Mathematical analysis|mathematical analysis]]: the creation of the theory of variational inequalities by [[Biography:Gaetano Fichera|Gaetano Fichera]] is described in §5, pages 282–284.
*{{Citation
  | last = Duvaut
  | first = Georges
  | contribution = Problèmes unilatéraux en mécanique des milieux continus
  | contribution-url = http://www.mathunion.org/ICM/ICM1970.3/Main/icm1970.3.0071.0078.ocr.pdf
  | series = ICM Proceedings
  | title = Actes du Congrès international des mathématiciens, 1970
  | volume = Mathématiques appliquées (E), Histoire et Enseignement (F) – Volume 3
  | pages = 71–78
  | year = 1971
  | place = [[Place:Paris|Paris]]
  | publisher = Gauthier-Villars
  | url = http://www.mathunion.org/ICM/ICM1970.3/
  | access-date = 2015-07-25
  | archive-url = https://web.archive.org/web/20150725141516/http://www.mathunion.org/ICM/ICM1970.3/
  | archive-date = 2015-07-25
  | url-status = dead
  }}. A brief research survey describing the field of variational inequalities, precisely the sub-field of [[Physics:Continuum mechanics|continuum mechanics]] problems with unilateral constraints.
*{{Citation
| first = Gaetano
| last = Fichera
| editor-last = 
| editor-first = 
| contribution = La nascita della teoria delle disequazioni variazionali ricordata dopo trent'anni 
| title = Incontro scientifico italo-spagnolo. Roma, 21 ottobre 1993
| url = http://www.lincei.it/pubblicazioni/catalogo/volume.php?lg=e&rid=32885
| language = Italian
| year = 1995
| pages = 47–53
| place = [[Place:Rome|Roma]]
| series = Atti dei Convegni Lincei
| volume = 114
| publisher = Accademia Nazionale dei Lincei
}}. ''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===
*{{Citation
| last1=Facchinei 
| first1=Francisco 
| last2=Pang 
| first2=Jong-Shi 
| title=Finite Dimensional Variational Inequalities and Complementarity Problems, Vol. 1
| series = Springer Series in Operations Research
| publisher=[[Physics:Springer-Verlag|Springer-Verlag]] 
| location= [[Place:Berlin|Berlin]]–Heidelberg–{{wipe|New York City}} 
| isbn=0-387-95580-1 
| year=2003
| zbl=1062.90001
}}
*{{Citation
| last1=Facchinei 
| first1=Francisco 
| last2=Pang 
| first2=Jong-Shi 
| title=Finite Dimensional Variational Inequalities and Complementarity Problems, Vol. 2
| series = Springer Series in Operations Research
| publisher=[[Physics:Springer-Verlag|Springer-Verlag]] 
| location= [[Place:Berlin|Berlin]]–Heidelberg–{{wipe|New York City}} 
| isbn=0-387-95581-X 
| year=2003
| zbl=1062.90001
}}
*{{citation
| last = Fichera
| first = Gaetano
| title = Sul problema elastostatico di Signorini con ambigue condizioni al contorno 
|trans-title = On the elastostatic problem of Signorini with ambiguous boundary conditions
| journal = Rendiconti della Accademia Nazionale dei Lincei, Classe di Scienze Fisiche, Matematiche e Naturali
| language = Italian
| volume = 34 
| series =  8
| issue = 2
| year = 1963
| url = http://www.bdim.eu/item?id=RLINA_1963_8_34_2_138_0| pages=138–142
| mr = 0176661
| zbl= 0128.18305
}}. A short research note announcing and describing (without proofs) the solution of the Signorini problem.
*{{citation
| last = Fichera
| first = Gaetano
| title = Problemi elastostatici con vincoli unilaterali: il problema di Signorini con ambigue condizioni al contorno
| trans-title = Elastostatic problems with unilateral constraints: the Signorini problem with ambiguous boundary conditions
| journal = Memorie della Accademia Nazionale dei Lincei, Classe di Scienze Fisiche, Matematiche e Naturali
| language = Italian
| volume = 7
| series =  8
| issue = 2
| year = 1964a
| pages=91–140
| zbl = 0146.21204
}}. The first paper where an [[Existence theorem|existence]] and [[Uniqueness theorem|uniqueness theorem]] for the Signorini problem is proved.
* {{citation
| last = Fichera
| first = Gaetano
| contribution = Elastostatic problems with unilateral constraints: the Signorini problem with ambiguous boundary conditions
| title = Seminari dell'istituto Nazionale di Alta Matematica 1962–1963
| year = 1964b
| publisher = Edizioni Cremonese
| place = [[Place:Rome|Rome]]
| pages=613–679
}}. An English translation of {{Harv|Fichera|1964a}}.
*{{Citation
| last1 = Glowinski
| first1 = Roland 
| last2 = Lions
| first2 = Jacques-Louis 
| last3 = Trémolières
| first3 = Raymond 
| title = Numerical analysis of variational inequalities. Translated from the French
| place = [[Place:Amsterdam|Amsterdam]]–{{wipe|New York City}}–Oxford
| publisher = [[Company:Elsevier|North-Holland]]
| year = 1981
| series = Studies in Mathematics and its Applications
| volume = 8
| mr = 635927 
| isbn = 0-444-86199-8
| zbl = 0463.65046
| pages = xxix+776 }}
*{{Citation
| last1=Kinderlehrer 
| first1=David 
| last2=Stampacchia 
| first2=Guido 
| title=An Introduction to Variational Inequalities and Their Applications 
| publisher=[[Company:Academic Press|Academic Press]] 
| series= Pure and Applied Mathematics
| url = https://books.google.com/books?id=B1cPRJ3qiw0C&q=An+Introduction+to+Variational+Inequalities+and+Their+Applications
| volume = 88
| location=Boston–[[Place:London|London]]–{{wipe|New York City}}–San Diego–[[Place:Sydney|Sydney]]–[[Place:Tokyo|Tokyo]]–Toronto
| isbn=0-89871-466-4
| year=1980
| zbl=0457.35001}}.
*{{Citation
| last1 = Lions
| first1 = Jacques-Louis
| last2 = Stampacchia
| first2 = Guido
| title = Inéquations variationnelles non coercives
| journal = Comptes rendus hebdomadaires des séances de l'Académie des sciences
| volume = 261
| pages = 25–27
| year = 1965
| url = http://gallica.bnf.fr/ark:/12148/bpt6k4022z.image.r=Comptes+Rendus+Academie.langEN.f26.pagination
| zbl = 0136.11906
}}, available at Gallica. Announcements of the results of paper {{Harv|Lions|Stampacchia|1967}}.
*{{Citation
| last1 = Lions
| first1 = Jacques-Louis
| last2 = Stampacchia
| first2 = Guido
| title = Variational inequalities
| journal = [[Communications on Pure and Applied Mathematics]]
| volume = 20
| pages = 493–519 
| year = 1967
| url = http://www3.interscience.wiley.com/journal/113397217/abstract
| archive-url = https://archive.today/20130105092629/http://www3.interscience.wiley.com/journal/113397217/abstract
| url-status = dead
| archive-date = 2013-01-05
| doi = 10.1002/cpa.3160200302
| zbl = 0152.34601
| issue = 3
}}. An important paper, describing the abstract approach of the authors to the theory of variational inequalities.
*{{citation
|last=Roubíček
|first= Tomáš
|title=Nonlinear Partial Differential Equations with Applications
|series=ISNM. International Series of Numerical Mathematics
|volume=153
|publisher= Birkhäuser Verlag
|place= Basel–Boston–Berlin
|pages= xx+476
|edition=2nd
|year= 2013
|isbn= 978-3-0348-0512-4
|mr=3014456
|zbl=1270.35005
|doi=10.1007/978-3-0348-0513-1}}.
*{{Citation
| last = Stampacchia
| first = Guido
| title = Formes bilineaires coercitives sur les ensembles convexes
| journal = Comptes rendus hebdomadaires des séances de l'Académie des sciences
| volume = 258
| pages = 4413–4416
| year = 1964
| url = http://gallica.bnf.fr/ark:/12148/bpt6k4012p.image.r=Comptes+Rendus+Academie.f20.langEN
| doi = 
| zbl = 0124.06401
}}, available at Gallica. The paper containing Stampacchia's generalization of the [[Lax–Milgram theorem]].

== External links ==
*{{springer
 | title= Variational inequalities
 | id= V/v120010
 | last= Panagiotopoulos
 | first= P.D. 
}}
* [https://www.scilag.net/problem/G-180630.1  Alessio Figalli,  On global homogeneous solutions to the Signorini problem],


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[[Category:Partial differential equations]]
[[Category:Calculus of variations]]

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