EQUILIBRIUM PROBLEMS

Antonino MAUGERI Dipartimento di Matematica e Informatica Università di Catania Viale A. Doria, 6 95125 Catania e-mail: maugeri@dmi.unict.it Tel.: +39 095 738 3059 Fax: +39 095 330094

Many equilibrium problems arising from various fields of science may be expressed, under general conditions, in a unified way:

\{ \begin{matrix}BuLu = 0 \\ Bu ≥ 0 Lu ≥ 0 (1) \\ u ∈ S \end{matrix}

where B and L are suitable operators defined in a suitable functional class S. The kind of EQUILIBRIUM described by the structure (1), in general is different from the one obtained by minimization of a cost functional or of an energy integral. Moreover, the structure (1) leads, in general, to a VARIATIONAL INEQUALITY on a convex, closed subset of S (usually called K). This happens, for example, for:

• Unilateral problems incontinuum mechanics (the celebrated Signorini Problem)

• the Obstacle Problem

• the Discrete and Continuum Traffic Equilibrium problem

• the Spatial Price Problem

• the Financial Problem

• the Walras problem

The equilibrium given by the structure (1) may be considered as an equilibrium from a ”local point of view”. It is different, in general, from the one, which we call ”global”, obtained by minimizing the usual functionals, and represents a complementary aspect which makes clear unknown features of equilibrium problems. The reason for which the past decades have witnessed an exceptional interest for the equilibrium problems of the type (1) rests on the fact that the Variational Inequality theory which in general expresses the equilibrium conditions (1), provides a powerful methodology, that in these last years has been improved by studying the connections with:

• The Separation Theory

• Gap Functions

• The Lagrangean Theory and Duality

• Many related Computational Procedures In the present paper some of the most recent results and some open problems are focused.