ESAIM: Control, Optimisation and Calculus of Variations

Table of Contents - October 2009 - Volume 15, Issue 04  

Research Article

A converse to the Lions-Stampacchia Theorem

Ernst, Emila1 and Théra, Michela2

a1 Aix-Marseille Univ, UMR6632, Marseille, 13397, France. Emil.Ernst@univ-cezanne.fr

a2 XLIM (UMR-CNRS ) and Université de Limoges, 123 Avenue A. Thomas, 87060 Limoges Cedex, France. michel.thera@unilim.fr

Abstract

In this paper we show that a linear variational inequality over an infinite dimensional real Hilbert space admits solutions for every nonempty bounded closed and convex set, if and only if the linear operator involved in the variational inequality is pseudo-monotone in the sense of Brezis.

(Received September 27 2007)

(Revised February 14 2008)

(Online publication August 20 2008)

Key Words:

  • Lions-Stampacchia Theorem;
  • variational inequality;
  • pseudo-monotone operator

Mathematics Subject Classification:

  • 47H05;
  • 52A41;
  • 39B82