Abstract
We study the mean curvature flow of radially symmetric graphs with prescribed contact angle on a fixed, smooth hypersurface in Euclidean space. In this paper we treat two distinct problems. The first problem has a free Neumann boundary only, while the second has two disjoint boundaries, a free Neumann boundary and a fixed Dirichlet height. We separate the two problems and prove that under certain initial conditions we have either long time existence followed by convergence to a minimal surface, or finite maximal time of existence at the end of which the graphs develop a curvature singularity. We also give a rate of convergence for the singularity.
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Acknowledgments
The results contained in this paper are part of the author’s Ph.D. thesis under the supervision of Prof. Klaus Ecker at the Free University in Berlin, in the group of Geometric Analysis as a Berlin Mathematical School student. The author would like to thank them for financial support and help during the completion of her thesis. This paper was written and completed during an extended visit to the University of Wollongong, Australia, as an invite of Prof. Graham Williams and Dr. James McCoy. The author is greatly indebted to them for their hospitality and advice during her stay, and would like to thank them for many enlightening discussions on this work. The author would also like to thank the anonymous referee whose careful reading and comments have led to an improvement of this paper.
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Wheeler, VM. Non-parametric radially symmetric mean curvature flow with a free boundary. Math. Z. 276, 281–298 (2014). https://doi.org/10.1007/s00209-013-1200-7
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DOI: https://doi.org/10.1007/s00209-013-1200-7