Variational inequalities with varying obstacles: The general form of the limit problem

When minimizing a functional of the calculus of variations with varying constraints of obstacle type, it may happen that, in the limit problem (when it exists), the constraint is no longer of obstacle type; it may take a “relaxed” form. This situation is analysed in this paper. Following De Giorgi's Γ-convergence approach, a class of functionals is constructed which is compact and which contains the obstacle constraint functionals. In this sense, this class of problems has been closed.