An existence result for optimal obstacles

We consider the optimization problem min{F(g): g is an element of X(Omega)} , where F(g) is a variational energy associated to the obstacle g and the c lass X(Omega) of admissible obstacles is given by X(Omega) = {g: Omega --> R: g less than or equal to psi on Omega, integral(Omega) g dx = c} with psi is an element of W-0(1,) (p)(Omega) and c is an element of R fixed. Genera lly, this problem does not have a solution and it may happen that the "opti mal" obstacle is of relaxed form. Under a monotonicity assumption on F, we prove the existence of a non-relaxed optimal obstacle in the family X(Omega ) through a new method based on the notions of y and wy-convergences. (C) 1 999 Academic Press.