Constrained optimal rearrangement problem leading to a free boundary problem with a non-local obstacle

We consider a new type of obstacle problem in the cylindrical domain, where the obstacle is imposed on the integral of the function with respect to the axis direction. We prove existence and regularity results and show that the comparison principle does not hold for the minimizers. This problem is derived from a classical optimal rearrangement problem in a cylindrical domain, under the constraint that the force function does not depend on the variable of the cylindrical axis. This leads to a new type of obstacle problem with a non-local obstacle.