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Variational inequality

释义

History
Definition
Examples
The problem of finding the minimal value of a real-valued function of real variable The general finite-dimensional variational inequality The variational inequality for the Signorini problem
See also
References
Historical references Scientific works
External links

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 {{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 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 {{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}}.

Definition

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

{{EquationRef|1|Definition 1.}} Given a Banach space

, a subset

, 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:

where

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 {{Harvtxt|Antman|1983|p=283}}: consider the problem of finding the minimal value of a differentiable function over a closed interval . Let be a point in where the minimum occurs. Three cases can occur:

if then
if then
if then

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 inequality

A formulation of the general problem in

is the following: given a subset

and a mapping

the finite-dimensional variational inequality problem associated with

consist of finding a -dimensional vector belonging to

such that

where

is the standard inner product on the vector space .

The variational inequality for the Signorini problem

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 elastic equilibrium configuration

of an anisotropic non-homogeneous elastic body that lies in a subset

of the three-dimensional euclidean space whose boundary is , resting on a rigid frictionless surface and subject only to its mass forces.

The solution of the problem exists and is unique (under precise assumptions) in the set of admissible displacements

i.e. the set of displacement vectors satisfying the system of ambiguous boundary conditions if and only if

where

and

are the following functionals,

written using the Einstein notation

, ,

where, for all ,

is the contact surface (or more generally a contact set),
is the body force applied to the body,
is the surface force applied to ,
is the infinitesimal strain tensor,
is the Cauchy stress tensor, defined as

where is the elastic potential energy and is the elasticity tensor.

References

Historical references

{{Citation

}}. 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.

{{Citation

}}. A brief research survey describing the field of variational inequalities, precisely the sub-field of continuum mechanics problems with unilateral constraints.

{{Citation

}}. 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
| 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

}}. "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.

{{citation

}}. "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.

{{citation

}}. An English translation of {{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 = 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 = https://books.google.com/books?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

}}, available at Gallica. Announcements of the results of paper {{Harv|Lions|Stampacchia|1967}}.

{{Citation

}}{{dead link|date=February 2019|bot=medic}}{{cbignore|bot=medic}}. 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

}}, 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=
}}

[https://www.scilag.net/problem/G-180630.1 Alessio Figalli, On global homogeneous solutions to the Signorini problem,]

2 : Partial differential equations|Calculus of variations

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History

Definition

Examples

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

The general finite-dimensional variational inequality

The variational inequality for the Signorini problem

See also

References

Historical references

Scientific works

External links