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arXiv · 1301.6322

Existence and Regularity for a Curvature Dependent Variational Problem

Jochen Denzler
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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.

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Jochen Denzler. 2013-01-27. Existence and Regularity for a Curvature Dependent Variational Problem. https://arxiv.org/abs/1301.6322

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