Generalizations
The theory of the obstacle problem is extended to other divergence form uniformly elliptic operators, and their associated energy functionals. It can be generalized to degenerate elliptic operators as well.
The double obstacle problem, where the function is constrained to lie above one obstacle function and below another, is also of interest.
The Signorini problem is a variant of the obstacle problem, where the energy functional is minimized subject to a constraint which only lives on a surface of one lesser dimension, which includes the boundary obstacle problem, where the constraint operates on the boundary of the domain.
The parabolic, time-dependent cases of the obstacle problem and its variants are also objects of study.
Read more about this topic: Obstacle Problem