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Existence and regularity for a curvature dependent variational problem


Author: Jochen Denzler
Journal: Trans. Amer. Math. Soc. 367 (2015), 3829-3845
MSC (2010): Primary 53A04; Secondary 49J45, 49N60, 49R50
Published electronically: December 11, 2014
MathSciNet review: 3324911
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Abstract: It is proved that smooth closed curves of given length minimizing the principal eigenvalue of the Schrödinger 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: geometric improvement algorithm, relaxed variational problem, asymptotic analysis, and strong variations.


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Additional Information

Jochen Denzler
Affiliation: Department of Mathematics, University of Tennessee, Knoxville, Tennessee 37996
Email: denzler@math.utk.edu

DOI: https://doi.org/10.1090/S0002-9947-2014-06188-6
Received by editor(s): December 7, 2012
Published electronically: December 11, 2014
Additional Notes: The author gratefully acknowledges repeated useful discussions with Almut Burchard. She inspired the research and was available to discuss and critique progress. Some of her ideas enter in the arguments, as outlined in the main text. This reseach was partly supported by a grant from the Simons Foundation (#208550). The hospitality of the CRM Université de Montréal during the workshop on Geometry of Eigenfunctions June 4-8, 2012 was a boost to this research, as was a Faculty Development Leave (‘Sabbatical’) granted by the University of Tennessee during Spring 2012, and the hospitality of Karlsruhe Institute of Technology during said leave.
Article copyright: © Copyright 2014 American Mathematical Society
The copyright for this article reverts to public domain 28 years after publication.