Existence and Regularity for a Curvature Dependent Variational Problem

It is proved that smooth closed curves of given length minimizing the principal eigenvalue of the Schrödinger operator − d 2 ds + κ exist. Here s denotes the arclength and κ the curvature. These minimizers are automatically planar, analytic, convex curves. The straight segment, traversed back and forth, is the only possible exception that becomes admissible in a more generalized setting. In proving this, we overcome the difficulty from a lack of coercivity and compactness by a combination of methods.