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| In [[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]]. The [[mathematical]] [[theory]] of variational inequalities was initially developed to deal with [[Equilibrium point|equilibrium]] problems, precisely the [[Signorini problem]]: in that model problem, the functional involved was obtained as the [[first variation]] of the involved [[Signorini_problem#The_potential_energy|potential energy]] therefore it has a [[Calculus of variation|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 (mathematics)|optimization]] and [[game theory]].
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| 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 {{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, [[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]]s and [[coin]]ed the name "variational inequality" for all the problems involving [[inequality (mathematics)|inequalities]] of this kind. [[Georges Duvaut]] encouraged his [[graduate student]]s 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 [[France]]. Also in 1965, Stampacchia and [[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}}.
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| == Definition == | | == Grey Burberry Scarf == |
| Following {{Harvtxt|Antman|1983|p=283}}, the formal definition of a variational inequality is the following one.
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| {{EquationRef|1|Definition 1.}} Given a [[Banach space]] '''<math>E</math>''', a [[subset]] '''<math>K</math>''' of '''<math>E</math>''', and a functional <math>\scriptstyle F:\boldsymbol{K}\rightarrow\boldsymbol{E}^\ast</math> from '''<math>K</math>''' to the [[dual space]] '''<math>E^*</math>''' of the space '''<math>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>K</math>''' the following [[inequality (mathematics)|inequality]]:
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| :<math>\langle F(x), y-x \rangle \geq 0\qquad\forall y \in \boldsymbol{K}</math>
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| where <math>\scriptstyle\langle\cdot,\cdot\rangle: \boldsymbol{E}^*\times\boldsymbol{E}\rightarrow\mathbb{R}</math> is the [[Dual space|duality pairing]].
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| In general, the variational inequality problem can be formulated on any [[Finite set|finite]] – or [[Infinite set|infinite]]-[[dimension]]al [[Banach space]]. The three obvious steps in the study of the problem are the following ones:
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| #Prove the existence of a solution: this step implies the ''mathematical correctness'' of the problem, showing that there is at least a solution.
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| #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.
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| #Find the solution.
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| ==Examples==
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| ===The problem of finding the minimal value of a real-valued function of real variable===
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| This is a standard example problem, reported by {{Harvtxt|Antman|1983|p=283}}: consider the problem of finding the [[minimum|minimal value]] of a [[differentiable function]] <math>f</math> over a [[closed interval]] <math>I = [a,b]</math>. Let <math>\scriptstyle x^*</math> be a point in ''<math>I</math>'' where the minimum occurs. Three cases can occur:
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| # if <math>\scriptstyle a<x^*< b</math> then <math>\scriptstyle f'(x^*) = 0;</math>
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| # if <math>\scriptstyle x^*=a</math> then <math>\scriptstyle f'(x^*) \ge 0;</math>
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| # if <math>\scriptstyle x^*=b</math> then <math>\scriptstyle f'(x^*) \le 0.</math>
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| These necessary conditions can be summarized as the problem of finding <math>\scriptstyle x^*\in I</math> such that
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| :<math>f'(x^*)(y-x^*) \geq 0\qquad\forall y \in I</math>
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| 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]], therefore this is a finite [[Dimension (mathematics)|dimensional]] variational inequality.
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| ===The general finite dimensional variational inequality===
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| A formulation of the general problem in <math>\scriptstyle\mathbb{R}^n</math> is the following: given a [[subset]] <math>K</math> of <math>\scriptstyle\mathbb{R}^n</math> and a [[Map (mathematics)|mapping]] <math>\scriptstyle F:K\rightarrow\mathbb{R}^n</math>, the [[Finite set|finite]]-[[dimension]]al variational inequality problem associated with <math>K</math> consist of finding a [[Dimension|<math>n</math>-dimensional]] [[Euclidean vector|vector]] <math>x</math> belonging to <math>K</math> such that
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| :<math>\langle F(x), y-x \rangle \geq 0\qquad\forall y \in K</math>
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| where <math>\scriptstyle\langle\cdot,\cdot\rangle:\mathbb{R}^n\times\mathbb{R}^n\rightarrow\mathbb{R}</math> is the standard [[inner product]] on the [[vector space]] <math>\scriptstyle\mathbb{R}^n</math>.
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| === The variational inequality for the Signorini problem ===
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| [[File:Classical Signorini problem.svg|thumb|right|400px|The classical [[Signorini problem]]: what will be the [[Linear elasticity#Elastostatics|equilibrium]] [[Continuum mechanics#Mathematical modeling of a continuum|configuration]] of the orange spherically shaped [[Physical body|elastic body]] resting on the blue [[Rigid body|rigid]] [[friction]]less [[Plane (geometry)|plane]]?]]
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| In the historical survey {{Harv|Fichera|1995}}, [[Gaetano Fichera]] describes the genesis of his solution to the [[Signorini problem]]: the problem consist in finding the [[Linear elasticity#Elastostatics|elastic equilibrium]] [[Continuum mechanics#Mathematical modeling of a continuum|configuration]] <math>\scriptstyle\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]] [[Homogeneous media|non-homogeneous]] [[Physical body|elastic body]] that lies in a [[subset]] <math>A</math> of the three-[[dimension]]al [[euclidean space]] whose [[boundary (topology)|boundary]] is <math>\scriptstyle\partial A</math>, resting on a [[Rigid body|rigid]] [[frictionless]] [[surface]] and subject only to its [[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>\scriptstyle\mathcal{U}_\Sigma</math> i.e. the set of [[displacement vector]]s satisfying the system of [[Signorini problem#The ambiguous boundary conditions|ambiguous boundary conditions]] if and only if
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| :<math>B(\boldsymbol{u},\boldsymbol{v}) - F(\boldsymbol{v}) \geq 0 \qquad \forall \boldsymbol{v} \in \mathcal{U}_\Sigma </math>
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| where <math>\scriptstyle B(\boldsymbol{u},\boldsymbol{v}) </math> and <math>\scriptstyle F(\boldsymbol{v}) </math> are the following [[Functional (mathematics)|functionals]], written using the [[Einstein notation]]
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| :<math>B(\boldsymbol{u},\boldsymbol{v}) = -\int_A \sigma_{ik}(\boldsymbol{u})\varepsilon_{ik}(\boldsymbol{v})\mathrm{d}x</math> {{spaces|6}} <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 \qquad \boldsymbol{u},\boldsymbol{v} \in \mathcal{U}_\Sigma </math>
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| where, for all <math>\scriptstyle\boldsymbol{x}\in A</math>,
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| *<math>\Sigma</math> is the [[Contact (mechanics)|contact]] [[surface]] (or more generally a contact [[set (mathematics)|set]]),
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| *<math>\scriptstyle\boldsymbol{f}(\boldsymbol{x}) = \left( f_1(\boldsymbol{x}), f_2(\boldsymbol{x}), f_3(\boldsymbol{x}) \right)</math> is the ''[[body force]]'' applied to the body,
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| *<math>\scriptstyle\boldsymbol{g}(\boldsymbol{x})=\left(g_1(\boldsymbol{x}),g_2(\boldsymbol{x}),g_3(\boldsymbol{x})\right)</math> is the [[surface force]] applied to <math>\scriptstyle\partial A\setminus\Sigma</math>,
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| *<math>\scriptstyle\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#Infinitesimal strain tensor|infinitesimal strain tensor]],
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| *<math>\scriptstyle\boldsymbol{\sigma}=\left(\sigma_{ik}\right)</math>is the [[Cauchy stress tensor]], defined as
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| ::<math>\sigma_{ik}= - \frac{\partial W}{\partial \varepsilon_{ik}} \qquad\forall i,k=1,2,3</math>
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| :where <math>\scriptstyle W(\boldsymbol{\varepsilon})=a_{ikjh}(\boldsymbol{x})\varepsilon_{ik}\varepsilon_{ik}</math> is the [[elastic potential energy]] and <math>\scriptstyle\boldsymbol{a}(\boldsymbol{x})=\left(a_{ikjh}(\boldsymbol{x})\right)</math> is the [[elasticity tensor]].
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| ==See also==
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| *[[Complementarity theory]]
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| *[[Differential variational inequality]]
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| *[[Mathematical programming with equilibrium constraints]]
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| *[[Obstacle problem]]
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| *[[Projected dynamical system]]
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| *[[Signorini problem]]
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| == Bibliography ==
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| *{{Citation
| |
| | last = Antman
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| | first = Stuart
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| | author-link =
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| | title = The influence of elasticity in analysis: modern developments
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| | journal = [[Bulletin of the American Mathematical Society]]
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| | volume = 9
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| | issue = 3
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| | pages = 267–291
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| | date =
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| | year = 1983
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| | url = http://www.ams.org/bull/1983-09-03/S0273-0979-1983-15185-6/home.html?pagingLink=%3Ca+href%3D%22%2Fjoursearch%2Fservlet%2FDoSearch%3Fco1%3Dand%26co2%3Dand%26co3%3Dand%26cperpage%3D50%26csort%3Dd%26endmo%3D00%26f1%3Dmsc%26f2%3Dtitle%26f3%3Danywhere%26f4%3Dauthor%26format%3Dstandard%26jrnl%3Done%26sendit22%3DSearch%26sperpage%3D30%26ssort%3Dd%26startmo%3D00%26timingString%3DQuery%2Btook%2B71%2Bmilliseconds.%26v4%3DAntman%26onejrnl%3Dbull%26startRec%3D1%22%3E
| |
| | doi = 10.1090/S0273-0979-1983-15185-6
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| | mr = 714990
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| | 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 paragraph 5, pages 282–284.
| |
| *{{Citation
| |
| | first = Gaetano
| |
| | last = Fichera
| |
| | author-link = Gaetano Fichera
| |
| | editor-last =
| |
| | editor-first =
| |
| | editor2-last =
| |
| | editor2-first =
| |
| | contribution = La nascita della teoria delle diseguaglianze variazionali ricordata dopo trent'anni (The birth of the theory of variational inequalities remembered thirty years later)
| |
| | contribution-url =
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| | title = Incontro scientifico italo-spagnolo. Roma, 21 ottobre 1993
| |
| | url = http://www.lincei.it/pubblicazioni/catalogo/volume.php?rid=36500
| |
| | year = 1995
| |
| | pages = 47–53
| |
| | place = [[Rome|Roma]]
| |
| | series = Atti dei Convegni Lincei
| |
| | volume = 114
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| | publisher = [[Accademia Nazionale dei Lincei]]
| |
| }} (in [[Italian language|Italian]]).
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| ==References==
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| *{{Citation
| |
| | last1=Facchinei
| |
| | first1=Francisco
| |
| | author1-link=
| |
| | last2=Pang
| |
| | first2=Jong-Shi
| |
| | author2-link=
| |
| | title=Finite Dimensional Variational Inequalities and Complementarity Problems, Vol. 1
| |
| | series = Springer Series in Operations Research
| |
| | publisher=[[Springer-Verlag]]
| |
| | location= [[Berlin]]-[[Heidelberg]]-[[New York]]
| |
| | isbn=0-387-95580-1
| |
| | year=2003
| |
| | zbl=1062.90001
| |
| }}
| |
| *{{Citation
| |
| | last1=Facchinei
| |
| | first1=Francisco
| |
| | author1-link=
| |
| | last2=Pang
| |
| | first2=Jong-Shi
| |
| | author2-link=
| |
| | title=Finite Dimensional Variational Inequalities and Complementarity Problems, Vol. 2
| |
| | series = Springer Series in Operations Research
| |
| | publisher=[[Springer-Verlag]]
| |
| | location= [[Berlin]]-[[Heidelberg]]-[[New York]]
| |
| | 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 (On the elastostatic problem of Signorini with ambiguous boundary conditions)
| |
| | journal = Rendiconti della [[Accademia Nazionale dei Lincei]], Classe di Scienze Fisiche, Matematiche e Naturali
| |
| | volume = 34
| |
| | series = 8
| |
| | issue = 2
| |
| | year = 1963
| |
| | pages=138–142
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| | zbl = 0128.18305
| |
| }} (in [[Italian language|Italian]]). A short paper describing briefly the approach to 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 (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
| |
| | volume = 7
| |
| | series = 8
| |
| | issue = 2
| |
| | year = 1964a
| |
| | pages=91–140
| |
| | zbl = 0146.21204
| |
| }} (in [[Italian language|Italian]]). The paper containing the [[Existence theorem|existence]] and [[uniqueness theorem]] for the [[Signorini problem]].
| |
| * {{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 = [[Rome]]
| |
| | pages=613–679
| |
| }}. An English translation of the paper {{Harv|Fichera|1964a}}.
| |
| *{{Citation
| |
| | last = Glowinski
| |
| | first = Roland
| |
| | author-link = Roland Glowinski
| |
| | last2 = Lions
| |
| | first2 = Jacques-Louis
| |
| | author2-link = Jacques-Louis Lions
| |
| | last3 = Trémolières
| |
| | first3 = Raymond
| |
| | author3-link=
| |
| | title = Numerical analysis of variational inequalities. Translated from the French
| |
| | place = [[Amsterdam]]-[[New York]]-[[Oxford]]
| |
| | publisher = [[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
| |
| | author1-link=David Kinderlehrer
| |
| | last2=Stampacchia
| |
| | first2=Guido
| |
| | author2-link=Guido Stampacchia
| |
| | title=An Introduction to Variational Inequalities and Their Applications
| |
| | publisher=[[Academic Press]]
| |
| | series= Pure and Applied Mathematics
| |
| | url = http://books.google.com/?id=B1cPRJ3qiw0C&printsec=frontcover&dq=An+Introduction+to+Variational+Inequalities+and+Their+Applications
| |
| | volume = 88
| |
| | location=[[Boston]]-[[London]]-[[New York]]-[[San Diego]]-[[Sydney]]-[[Tokyo]]-[[Toronto]]
| |
| | isbn=0-89871-466-4
| |
| | year=1980
| |
| | zbl=0457.35001}}.
| |
| *{{Citation
| |
| | last = Lions
| |
| | first = Jacques-Louis
| |
| | author-link = Jacques-Louis Lions
| |
| | last2 = Stampacchia
| |
| | first2 = Guido
| |
| | author2-link = Guido Stampacchia
| |
| | 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
| |
| | last = Lions
| |
| | first = Jacques-Louis
| |
| | author-link = Jacques-Louis Lions
| |
| | last2 = Stampacchia
| |
| | first2 = Guido
| |
| | author2-link = Guido Stampacchia
| |
| | title = Variational inequalities
| |
| | journal = [http://www3.interscience.wiley.com/journal/29240/home?CRETRY=1&SRETRY=0 Communications on Pure and Applied Mathematics]
| |
| | volume = 20
| |
| | pages = 493–519
| |
| | year = 1967
| |
| | url = http://www3.interscience.wiley.com/journal/113397217/abstract
| |
| | 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 = Stampacchia
| |
| | first = Guido
| |
| | author-link = Guido Stampacchia
| |
| | 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.
| |
| | author-link=
| |
| }}
| |
| | |
| {{DEFAULTSORT:Variational Inequality}}
| |
| [[Category:Partial differential equations]]
| |
| [[Category:Calculus of variations]]
| |
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