Existence and regularity for a curvature dependent variational problem
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- by Jochen Denzler PDF
- Trans. Amer. Math. Soc. 367 (2015), 3829-3845 Request permission
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.References
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Additional Information
- Jochen Denzler
- Affiliation: Department of Mathematics, University of Tennessee, Knoxville, Tennessee 37996
- MR Author ID: 250152
- Email: denzler@math.utk.edu
- 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.
- © Copyright 2014
American Mathematical Society
The copyright for this article reverts to public domain 28 years after publication. - Journal: Trans. Amer. Math. Soc. 367 (2015), 3829-3845
- MSC (2010): Primary 53A04; Secondary 49J45, 49N60, 49R50
- DOI: https://doi.org/10.1090/S0002-9947-2014-06188-6
- MathSciNet review: 3324911