Abstract
Let S be a smooth hypersurface properly embedded in \(\mathbb R^N\) with \(N \ge 3\) and consider its tubular neighborhood \(\mathscr {N}\). We show that, if a heat flow over \(\mathscr {N}\) with appropriate initial and boundary conditions has S as a stationary isothermic surface, then S must have some sort of symmetry.
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Acknowledgments
This research was partially supported by the Grant-in-Aid for Challenging Exploratory Research (\(\sharp \) 25610024) of Japan Society for the Promotion of Science. The author would like to thank the anonymous referees for their some valuable suggestions to improve the presentation and clarity in several points.
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Sakaguchi, S. (2016). Symmetry Problems on Stationary Isothermic Surfaces in Euclidean Spaces. In: Gazzola, F., Ishige, K., Nitsch, C., Salani, P. (eds) Geometric Properties for Parabolic and Elliptic PDE's. GPPEPDEs 2015. Springer Proceedings in Mathematics & Statistics, vol 176. Springer, Cham. https://doi.org/10.1007/978-3-319-41538-3_13
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DOI: https://doi.org/10.1007/978-3-319-41538-3_13
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