TOPICS ON VARIATIONAL ANALYSIS AND APPLICATIONS TO EQUILIBRIUM PROBLEMS

Antonino MAUGERI Dipartimento di Matematica e Informatica, Università di Catania Viale A. Doria, 6 – 95125 CATANIA (ITALIA)

                    e-mail: maugeri@dmi.unict.it

Abstract. An important role plays a broad class of equilibrium problems, whose equilibrium conditions are formalized by the relation

\{ \begin{matrix}B(u)L(u) = 0 \\ B(u) ≥ 0, L(u) ≥ 0 \\ u ∈ S \end{matrix}

where S is a suitable functional space. The previous relation usually leads to a Variational (or Quasi - Variational) Inequality on a convex closed subset of S or, in particular cases, to a Complementarity Problem. Examples of such a formulation, that have been already reached or are still under consideration, are the discrete and continuum traffic equilibrium problem, the price equilibrium problem in a spatial distributed market, the Walras problem, the Migration problem, the obstacle problem, Signorini’s problem, the elastic–plastic torsion problem.

As a particular case of variational analysis, the Variational Inequalities, together with their connected concepts of Duality, Lagrangean Theory, Gap Function, Minimax Theory, provide a powerful methodology that allows to describe important features of problems whose equilibrium conditions are given by the structure above.

It is worth mentioning that when we pass from the finite dimensional case to the infinite dimensional one, we need new appropriate concepts that allow to apply the dual theory and to find Lagrangean multipliers associated to Variational Inequalities. Examples of such new concepts arise when we consider applications to the elastic–plastic torsion problems, for which we can solve the problem of finding the Lagrangean multipliers or the continuum traffic equilibrium problem or evolutionary Variational Inequalities.