Existence and Regularity for a Curvature Dependent Variational Problem

2013/01/27 by Jochen Denzler, Denzler, Jochen
Engineering · Mathematics · #49J45 #49N60 #49R50 #53A04 #FOS: Physical sciences #Mathematical Physics (math-ph) #Numerical methods in engineering #Numerical methods in inverse problems #Spectral Theory in Mathematical Physics

paper · pdf · doi:10.48550/arxiv.1301.6322

openalex publication_date 2013/01/27 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28

Abstract

It is proved that smooth closed curves of given length minimizing the principal eigenvalue of the Schrödinger operator -(d²)/(ds²)+κ² 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.

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