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mathematics Mathematics is an area of knowledge that includes the topics of numbers, formulas and related structures, shapes and the spaces in which they are contained, and quantities and their changes. These topics are represented in modern mathematics ...
, a variational inequality is an inequality involving a
functional Functional may refer to: * Movements in architecture: ** Functionalism (architecture) ** Form follows function * Functional group, combination of atoms within molecules * Medical conditions without currently visible organic basis: ** Functional sy ...
, which has to be solved for all possible values of a given variable, belonging usually to a
convex set In geometry, a subset of a Euclidean space, or more generally an affine space over the reals, is convex if, given any two points in the subset, the subset contains the whole line segment that joins them. Equivalently, a convex set or a convex ...
. The
mathematical Mathematics is an area of knowledge that includes the topics of numbers, formulas and related structures, shapes and the spaces in which they are contained, and quantities and their changes. These topics are represented in modern mathematics ...
theory A theory is a rational type of abstract thinking about a phenomenon, or the results of such thinking. The process of contemplative and rational thinking is often associated with such processes as observational study or research. Theories may ...
of variational inequalities was initially developed to deal with equilibrium problems, precisely the
Signorini problem The Signorini problem is an elastostatics problem in linear elasticity: it consists in finding the elastic equilibrium configuration of an anisotropic non-homogeneous elastic body, resting on a rigid frictionless surface and subject only to i ...
: in that model problem, the functional involved was obtained as the
first variation In applied mathematics and the calculus of variations, the first variation of a functional ''J''(''y'') is defined as the linear functional \delta J(y) mapping the function ''h'' to :\delta J(y,h) = \lim_ \frac = \left.\frac J(y + \varepsilon h ...
of the involved
potential energy In physics, potential energy is the energy held by an object because of its position relative to other objects, stresses within itself, its electric charge, or other factors. Common types of potential energy include the gravitational potenti ...
. 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 Economics () is the social science that studies the production, distribution, and consumption of goods and services. Economics focuses on the behaviour and interactions of economic agents and how economies work. Microeconomics anal ...
,
finance Finance is the study and discipline of money, currency and capital assets. It is related to, but not synonymous with economics, the study of production, distribution, and consumption of money, assets, goods and services (the discipline of f ...
,
optimization Mathematical optimization (alternatively spelled ''optimisation'') or mathematical programming is the selection of a best element, with regard to some criterion, from some set of available alternatives. It is generally divided into two subfi ...
and
game theory Game theory is the study of mathematical models of strategic interactions among rational agents. Myerson, Roger B. (1991). ''Game Theory: Analysis of Conflict,'' Harvard University Press, p.&nbs1 Chapter-preview links, ppvii–xi It has appli ...
.


History

The first problem involving a variational inequality was the
Signorini problem The Signorini problem is an elastostatics problem in linear elasticity: it consists in finding the elastic equilibrium configuration of an anisotropic non-homogeneous elastic body, resting on a rigid frictionless surface and subject only to i ...
, posed by Antonio Signorini in 1959 and solved by
Gaetano Fichera Gaetano Fichera (8 February 1922 – 1 June 1996) was an Italian mathematician, working in mathematical analysis, linear elasticity, partial differential equations and several complex variables. He was born in Acireale, and died in Rome. Biog ...
in 1963, according to the references and : the first papers of the theory were and , . Later on,
Guido Stampacchia Guido Stampacchia (26 March 1922 – 27 April 1978) was an Italian mathematician, known for his work on the theory of variational inequalities, the calculus of variation and the theory of elliptic partial differential equations.. Life and academ ...
proved his generalization to the
Lax–Milgram theorem Weak formulations are important tools for the analysis of mathematical equations that permit the transfer of concepts of linear algebra to solve problems in other fields such as partial differential equations. In a weak formulation, equations or c ...
in in order to study the regularity problem for
partial differential equation In mathematics, a partial differential equation (PDE) is an equation which imposes relations between the various partial derivatives of a multivariable function. The function is often thought of as an "unknown" to be solved for, similarly to h ...
s and
coin A coin is a small, flat (usually depending on the country or value), round piece of metal or plastic used primarily as a medium of exchange or legal tender. They are standardized in weight, and produced in large quantities at a mint in order ...
ed the name "variational inequality" for all the problems involving
inequalities Inequality may refer to: Economics * Attention inequality, unequal distribution of attention across users, groups of people, issues in etc. in attention economy * Economic inequality, difference in economic well-being between population groups * ...
of this kind.
Georges Duvaut Georges may refer to: Places *Georges River, New South Wales, Australia *Georges Quay (Dublin) *Georges Township, Fayette County, Pennsylvania Other uses *Georges (name) * ''Georges'' (novel), a novel by Alexandre Dumas * "Georges" (song), a 1977 ...
encouraged his
graduate student Postgraduate or graduate education refers to academic or professional degrees, certificates, diplomas, or other qualifications pursued by post-secondary students who have earned an undergraduate (bachelor's) degree. The organization and s ...
s to study and expand on Fichera's work, after attending a conference in
Brixen Brixen (, ; it, Bressanone ; lld, Porsenù or ) is a town in South Tyrol, northern Italy, located about north of Bolzano. Geography First mentioned in 901, Brixen is the third largest city and oldest town in the province, and the artistic an ...
on 1965 where Fichera presented his study of the Signorini problem, as reports: thus the theory become widely known throughout
France France (), officially the French Republic ( ), is a country primarily located in Western Europe. It also comprises of Overseas France, overseas regions and territories in the Americas and the Atlantic Ocean, Atlantic, Pacific Ocean, Pac ...
. Also in 1965, Stampacchia and
Jacques-Louis Lions Jacques-Louis Lions (; 3 May 1928 – 17 May 2001) was a French mathematician who made contributions to the theory of partial differential equations and to stochastic control, among other areas. He received the SIAM's John von Neumann Lecture ...
extended earlier results of , announcing them in the paper : full proofs of their results appeared later in the paper .


Definition

Following , the definition of a variational inequality is the following one. Given a
Banach space In mathematics, more specifically in functional analysis, a Banach space (pronounced ) is a complete normed vector space. Thus, a Banach space is a vector space with a metric that allows the computation of vector length and distance between vector ...
\boldsymbol, a
subset In mathematics, set ''A'' is a subset of a set ''B'' if all elements of ''A'' are also elements of ''B''; ''B'' is then a superset of ''A''. It is possible for ''A'' and ''B'' to be equal; if they are unequal, then ''A'' is a proper subset of ...
\boldsymbol of \boldsymbol, and a functional F\colon \boldsymbol\to \boldsymbol^ from \boldsymbol to the
dual space In mathematics, any vector space ''V'' has a corresponding dual vector space (or just dual space for short) consisting of all linear forms on ''V'', together with the vector space structure of pointwise addition and scalar multiplication by cons ...
\boldsymbol^ of the space \boldsymbol, the variational inequality problem is the problem of solving for the variable x belonging to \boldsymbol the following inequality: :\langle F(x), y-x \rangle \geq 0\qquad\forall y \in \boldsymbol where \langle\cdot,\cdot\rangle\colon \boldsymbol^\times\boldsymbol\to \mathbb is the
duality pairing Duality may refer to: Mathematics * Duality (mathematics), a mathematical concept ** Dual (category theory), a formalization of mathematical duality ** Duality (optimization) ** Duality (order theory), a concept regarding binary relations ** D ...
. In general, the variational inequality problem can be formulated on any
finite Finite is the opposite of infinite. It may refer to: * Finite number (disambiguation) * Finite set, a set whose cardinality (number of elements) is some natural number * Finite verb, a verb form that has a subject, usually being inflected or marke ...
– or
infinite Infinite may refer to: Mathematics * Infinite set, a set that is not a finite set *Infinity, an abstract concept describing something without any limit Music *Infinite (group), a South Korean boy band *''Infinite'' (EP), debut EP of American m ...
-
dimension In physics and mathematics, the dimension of a mathematical space (or object) is informally defined as the minimum number of coordinates needed to specify any point within it. Thus, a line has a dimension of one (1D) because only one coord ...
al
Banach space In mathematics, more specifically in functional analysis, a Banach space (pronounced ) is a complete normed vector space. Thus, a Banach space is a vector space with a metric that allows the computation of vector length and distance between vector ...
. 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 : consider the problem of finding the minimal value of a
differentiable function In mathematics, a differentiable function of one real variable is a function whose derivative exists at each point in its domain. In other words, the graph of a differentiable function has a non- vertical tangent line at each interior point in ...
f over a
closed interval In mathematics, a (real) interval is a set of real numbers that contains all real numbers lying between any two numbers of the set. For example, the set of numbers satisfying is an interval which contains , , and all numbers in between. Other ...
I = ,b/math>. Let x^ be a point in I where the minimum occurs. Three cases can occur: # if a then f^(x^) = 0; # if x^=a, then f^(x^) \ge 0; # if x^=b, then f^(x^) \le 0. These necessary conditions can be summarized as the problem of finding x^\in I such that :f^(x^)(y-x^) \geq 0\quad for \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 In mathematics, a real number is a number that can be used to measure a ''continuous'' one-dimensional quantity such as a distance, duration or temperature. Here, ''continuous'' means that values can have arbitrarily small variations. Every ...
, therefore this is a finite
dimensional In physics and mathematics, the dimension of a mathematical space (or object) is informally defined as the minimum number of coordinates needed to specify any point within it. Thus, a line has a dimension of one (1D) because only one coordi ...
variational inequality.


The general finite-dimensional variational inequality

A formulation of the general problem in \mathbb^n is the following: given a
subset In mathematics, set ''A'' is a subset of a set ''B'' if all elements of ''A'' are also elements of ''B''; ''B'' is then a superset of ''A''. It is possible for ''A'' and ''B'' to be equal; if they are unequal, then ''A'' is a proper subset of ...
K of \mathbb^ and a mapping F\colon K\to\mathbb^, the
finite Finite is the opposite of infinite. It may refer to: * Finite number (disambiguation) * Finite set, a set whose cardinality (number of elements) is some natural number * Finite verb, a verb form that has a subject, usually being inflected or marke ...
-
dimension In physics and mathematics, the dimension of a mathematical space (or object) is informally defined as the minimum number of coordinates needed to specify any point within it. Thus, a line has a dimension of one (1D) because only one coord ...
al variational inequality problem associated with K consist of finding a n-dimensional
vector Vector most often refers to: *Euclidean vector, a quantity with a magnitude and a direction *Vector (epidemiology), an agent that carries and transmits an infectious pathogen into another living organism Vector may also refer to: Mathematic ...
x belonging to K such that :\langle F(x), y-x \rangle \geq 0\qquad\forall y \in K where \langle\cdot,\cdot\rangle\colon\mathbb^\times\mathbb^\to\mathbb is the standard
inner product In mathematics, an inner product space (or, rarely, a Hausdorff pre-Hilbert space) is a real vector space or a complex vector space with an operation called an inner product. The inner product of two vectors in the space is a scalar, often ...
on the
vector space In mathematics and physics, a vector space (also called a linear space) is a set whose elements, often called '' vectors'', may be added together and multiplied ("scaled") by numbers called ''scalars''. Scalars are often real numbers, but can ...
\mathbb^.


The variational inequality for the Signorini problem

In the historical survey ,
Gaetano Fichera Gaetano Fichera (8 February 1922 – 1 June 1996) was an Italian mathematician, working in mathematical analysis, linear elasticity, partial differential equations and several complex variables. He was born in Acireale, and died in Rome. Biog ...
describes the genesis of his solution to the
Signorini problem The Signorini problem is an elastostatics problem in linear elasticity: it consists in finding the elastic equilibrium configuration of an anisotropic non-homogeneous elastic body, resting on a rigid frictionless surface and subject only to i ...
: the problem consist in finding the elastic equilibrium
configuration Configuration or configurations may refer to: Computing * Computer configuration or system configuration * Configuration file, a software file used to configure the initial settings for a computer program * Configurator, also known as choice bo ...
\boldsymbol(\boldsymbol) =\left(u_1(\boldsymbol),u_2(\boldsymbol),u_3(\boldsymbol)\right) of an
anisotropic Anisotropy () is the property of a material which allows it to change or assume different properties in different directions, as opposed to isotropy. It can be defined as a difference, when measured along different axes, in a material's physic ...
non-homogeneous elastic body that lies in a
subset In mathematics, set ''A'' is a subset of a set ''B'' if all elements of ''A'' are also elements of ''B''; ''B'' is then a superset of ''A''. It is possible for ''A'' and ''B'' to be equal; if they are unequal, then ''A'' is a proper subset of ...
A of the three-
dimension In physics and mathematics, the dimension of a mathematical space (or object) is informally defined as the minimum number of coordinates needed to specify any point within it. Thus, a line has a dimension of one (1D) because only one coord ...
al
euclidean space Euclidean space is the fundamental space of geometry, intended to represent physical space. Originally, that is, in Euclid's ''Elements'', it was the three-dimensional space of Euclidean geometry, but in modern mathematics there are Euclidea ...
whose boundary is \partial A, resting on a rigid
frictionless Frictionless can refer to: * Frictionless market * Frictionless continuant Approximants are speech sounds that involve the articulators approaching each other but not narrowly enough nor with enough articulatory precision to create turbulent ...
surface A surface, as the term is most generally used, is the outermost or uppermost layer of a physical object or space. It is the portion or region of the object that can first be perceived by an observer using the senses of sight and touch, and is ...
and subject only to its mass forces. The solution u of the problem exists and is unique (under precise assumptions) in the
set Set, The Set, SET or SETS may refer to: Science, technology, and mathematics Mathematics *Set (mathematics), a collection of elements *Category of sets, the category whose objects and morphisms are sets and total functions, respectively Electro ...
of admissible displacements \mathcal_\Sigma i.e. the set of displacement vectors satisfying the system of ambiguous boundary conditions if and only if :B(\boldsymbol,\boldsymbol - \boldsymbol) - F(\boldsymbol - \boldsymbol) \geq 0 \qquad \forall \boldsymbol \in \mathcal_\Sigma where B(\boldsymbol,\boldsymbol) and F(\boldsymbol) are the following functionals, written using the
Einstein notation In mathematics, especially the usage of linear algebra in Mathematical physics, Einstein notation (also known as the Einstein summation convention or Einstein summation notation) is a notational convention that implies summation over a set of ...
:B(\boldsymbol,\boldsymbol) = -\int_A \sigma_(\boldsymbol)\varepsilon_(\boldsymbol)\,\mathrmx,    F(\boldsymbol) = \int_A v_i f_i\,\mathrmx + \int_\!\!\!\!\! v_i g_i \,\mathrm\sigma,    \boldsymbol,\boldsymbol \in \mathcal_\Sigma where, for all \boldsymbol\in A, *\Sigma is the
contact Contact may refer to: Interaction Physical interaction * Contact (geology), a common geological feature * Contact lens or contact, a lens placed on the eye * Contact sport, a sport in which players make contact with other players or objects * C ...
surface A surface, as the term is most generally used, is the outermost or uppermost layer of a physical object or space. It is the portion or region of the object that can first be perceived by an observer using the senses of sight and touch, and is ...
(or more generally a contact
set Set, The Set, SET or SETS may refer to: Science, technology, and mathematics Mathematics *Set (mathematics), a collection of elements *Category of sets, the category whose objects and morphisms are sets and total functions, respectively Electro ...
), *\boldsymbol(\boldsymbol) = \left( f_1(\boldsymbol), f_2(\boldsymbol), f_3(\boldsymbol) \right) is the ''
body force In physics, a body force is a force that acts throughout the volume of a body. Springer site - Book 'Solid mechanics'preview paragraph 'Body forces'./ref> Forces due to gravity, electric fields and magnetic fields are examples of body forces. ...
'' applied to the body, *\boldsymbol(\boldsymbol)=\left(g_1(\boldsymbol),g_2(\boldsymbol),g_3(\boldsymbol)\right) is the
surface force Surface force denoted ''fs'' is the force that acts across an internal or external surface element in a material body. Surface force can be decomposed into two perpendicular components: normal forces and shear forces. A normal force acts normal ...
applied to \partial A\!\setminus\!\Sigma, *\boldsymbol=\boldsymbol(\boldsymbol)=\left(\varepsilon_(\boldsymbol)\right)=\left(\frac \left( \frac + \frac \right)\right) is the infinitesimal strain tensor, *\boldsymbol=\left(\sigma_\right) is the
Cauchy stress tensor In continuum mechanics, the Cauchy stress tensor \boldsymbol\sigma, true stress tensor, or simply called the stress tensor is a second order tensor named after Augustin-Louis Cauchy. The tensor consists of nine components \sigma_ that completely ...
, defined as ::\sigma_= - \frac \qquad\forall i,k=1,2,3 :where W(\boldsymbol)=a_(\boldsymbol)\varepsilon_\varepsilon_ is the
elastic potential energy Elastic energy is the mechanical potential energy stored in the configuration of a material or physical system as it is subjected to elastic deformation by work performed upon it. Elastic energy occurs when objects are impermanently compressed, ...
and \boldsymbol(\boldsymbol)=\left(a_(\boldsymbol)\right) is the
elasticity tensor In physics, Hooke's law is an empirical law which states that the force () needed to extend or compress a spring by some distance () scales linearly with respect to that distance—that is, where is a constant factor characteristic of ...
.


See also

*
Complementarity theory A complementarity problem is a type of mathematical optimization problem. It is the problem of optimizing (minimizing or maximizing) a function of two vector variables subject to certain requirements (constraints) which include: that the inner ...
*
Differential variational inequality In mathematics, a differential variational inequality (DVI) is a dynamical system that incorporates ordinary differential equations and variational inequalities or complementarity problems. DVIs are useful for representing models involving both ...
* Extended Mathematical Programming for Equilibrium Problems *
Mathematical programming with equilibrium constraints Mathematical programming with equilibrium constraints (MPEC) is the study of constrained optimization problems where the constraints include variational inequalities or complementarities. MPEC is related to the Stackelberg game. MPEC is used ...
*
Obstacle problem The obstacle problem is a classic motivating example in the mathematical study of variational inequalities and free boundary problems. The problem is to find the equilibrium position of an elastic membrane whose boundary is held fixed, and which i ...
*
Projected dynamical system Projected dynamical systems is a mathematical theory investigating the behaviour of dynamical systems where solutions are restricted to a constraint set. The discipline shares connections to and applications with both the static world of optimizatio ...
*
Signorini problem The Signorini problem is an elastostatics problem in linear elasticity: it consists in finding the elastic equilibrium configuration of an anisotropic non-homogeneous elastic body, resting on a rigid frictionless surface and subject only to i ...
*
Unilateral contact In contact mechanics, the term unilateral contact, also called unilateral constraint, denotes a mechanical constraint which prevents penetration between two rigid/flexible bodies. Constraints of this kind are omnipresent in non-smooth multibody d ...


References


Historical references

*. An historical paper about the fruitful interaction of elasticity theory and
mathematical analysis Analysis is the branch of mathematics dealing with continuous functions, limits, and related theories, such as differentiation, integration, measure, infinite sequences, series, and analytic functions. These theories are usually studied ...
: the creation of the theory of variational inequalities by
Gaetano Fichera Gaetano Fichera (8 February 1922 – 1 June 1996) was an Italian mathematician, working in mathematical analysis, linear elasticity, partial differential equations and several complex variables. He was born in Acireale, and died in Rome. Biog ...
is described in §5, pages 282–284. *. A brief research survey describing the field of variational inequalities, precisely the sub-field of
continuum mechanics Continuum mechanics is a branch of mechanics that deals with the mechanical behavior of materials modeled as a continuous mass rather than as discrete particles. The French mathematician Augustin-Louis Cauchy was the first to formulate such mo ...
problems with unilateral constraints. *. ''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

* * *. A short research note announcing and describing (without proofs) the solution of the Signorini problem. *. The first paper where an
existence Existence is the ability of an entity to interact with reality. In philosophy, it refers to the ontological property of being. Etymology The term ''existence'' comes from Old French ''existence'', from Medieval Latin ''existentia/exsistentia' ...
and
uniqueness theorem In mathematics, a uniqueness theorem, also called a unicity theorem, is a theorem asserting the uniqueness of an object satisfying certain conditions, or the equivalence of all objects satisfying the said conditions. Examples of uniqueness theorems ...
for the Signorini problem is proved. * . An English translation of . * *. *, available at Gallica. Announcements of the results of paper . *. An important paper, describing the abstract approach of the authors to the theory of variational inequalities. *. *, available at Gallica. The paper containing Stampacchia's generalization of the
Lax–Milgram theorem Weak formulations are important tools for the analysis of mathematical equations that permit the transfer of concepts of linear algebra to solve problems in other fields such as partial differential equations. In a weak formulation, equations or c ...
.


External links

*
Alessio Figalli, On global homogeneous solutions to the Signorini problem
{{DEFAULTSORT:Variational Inequality Partial differential equations Calculus of variations