Existence and Regularity for a Curvature Dependent Variational Problem

TL;DR

This paper proves the existence and regularity of smooth, convex, planar curves that minimize the principal eigenvalue of a curvature-dependent Schrödinger operator, with special considerations for non-coercive settings.

Contribution

It establishes the existence and regularity of minimizers for a curvature-dependent eigenvalue problem, including handling non-coercivity issues.

Findings

Minimizers are smooth, convex, planar, and analytic curves.

Straight segments are admissible in a generalized setting.

Existence is proven despite lack of coercivity and compactness.

Abstract

It is proved that smooth closed curves of given length minimizing the principal eigenvalue of the Schr\"odinger operator $-\frac{d^2}{ds^2}+\kappa^2$ exist. Here $s$ denotes the arclength and $\kappa$ 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.