Abstract
Motivated by homothetic solutions in curvature-driven flows of planar curves, as well as their many physical applications, this article carries out a systematic study of oriented smooth curves whose curvature \(\kappa \) is a given function of position or direction. The analysis is informed by a dynamical systems point of view. Though focussed on situations where the prescribed curvature depends only on the distance r from one distinguished point, the basic dynamical concepts are seen to be applicable in other situations as well. As an application, a complete classification of all closed solutions of \(\kappa = ar^{b}\), with arbitrary real constants \(a,b\), is established.
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Acknowledgements
The author was partially supported by an Nserc Discovery Grant. He owes deep gratitude to T.P. Hill and K.E. Morrison who in 2005 conjectured (correctly, as it turned out) that the only Jordan solution of \(\kappa = r\) is the (counter-clockwise oriented) unit circle, and who greatly helped this work come to fruition through continued interest and advice. Insightful comments by an anonymous referee led to a much improved presentation. Thanks also to J. Muldowney, M. Niksirat, T. Schmah, and C. Xu for several enlightening conversations over the years. Parts of this work were completed while the author was a visitor at the Universität Wien. He is much indebted to R. Zweimüller for many kind acts of hospitality.
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Berger, A. On Planar Curves with Position-Dependent Curvature. J Dyn Diff Equat 36, 975–1014 (2024). https://doi.org/10.1007/s10884-022-10168-9
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DOI: https://doi.org/10.1007/s10884-022-10168-9