An Existence Result for Optimal Obstacles

We consider the optimization problem min{F(g):g∈X(Ω)}, whereF(g) is a variational energy associated to the obstaclegand the classX(Ω) of admissible obstacles is given byX(Ω)={g: Ω→R: g⩽ψonΩ, ∫Ω g dx=c} withψ∈W1, p0(Ω) andc∈R fixed. Generally, this problem does not have a solution and it may happen that the “optimal” obstacle is of relaxed form. Under a monotonicity assumption onF, we prove the existence of a non-relaxed optimal obstacle in the familyX(Ω) through a new method based on the notions ofγandwγ-convergences.