Variational inequality


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 and : the first papers of the theory were and,. Later on, Guido Stampacchia proved his generalization to the Lax–Milgram theorem in 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 reports: thus the theory become widely known throughout France. Also in 1965, Stampacchia and Jacques-Louis Lions extended earlier results of, announcing them in the paper : full proofs of their results appeared later in the paper.

Definition

Following, the formal definition of a variational inequality is the following one.
Given a Banach space, a subset of, 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:
  1. Prove the existence of a solution: this step implies the mathematical correctness of the problem, showing that there is at least a solution.
  2. 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.
  3. 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 : 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:
  1. if then
  2. if then
  3. if then
These necessary conditions can be summarized as the problem of finding such that
The absolute minimum must be searched between the solutions 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
of
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, 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 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,