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

Planar Infinite-Horizon Optimal Control Problems with Weighted Average Cost and Constraints, Applied to Cheeger Sets