Existence of Minima for Nonconvex Functionals in Spaces of Functions Depending on the Distance from the Boundary

We prove that some nonconvex functionals admit a unique minimum in a func-tional space of functions which depend only on the distance from the boundary of the (plane) domain where they are defined. The domains considered are disks and regular polygons. We prove that the sequence of minima of the functional on the polygons converges to the unique minimum on the circumscribed disk as the num-ber of sides tends to infinity. Our method also allows us to determine the explicit form of the minima. 1