  We establish a Poincar\`E-Bendixson type result for a weighted averaged
infinite horizon problem in the plane, with and without averaged constraints.
For the unconstrained problem, we establish the existence of a periodic optimal
solution, and for constrained problem, we establish the existence of an optimal
solution that alternates cyclicly between a finite number of periodic curves,
depending on the number of constraints. Applications of these results are
presented to the shape optimization problems of the Cheeger set and the
generalized Cheeger set, and also to a singular limit of the one-dimensional
Cahn-Hilliard equation
