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. 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 . Following , the 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.