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On the regularity of the Neumann problem for free surfaces with surface tension


Authors: Walter Craig and Ana-Maria Matei
Journal: Proc. Amer. Math. Soc. 135 (2007), 2497-2504
MSC (2000): Primary 35J65, 76B15
DOI: https://doi.org/10.1090/S0002-9939-07-08776-X
Published electronically: March 14, 2007
MathSciNet review: 2302570
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Abstract: In 1952 H. Lewy established that a hydrodynamic free surface which is at least $ C^1$ in a neighborhood of a point $ q$ situated on the free surface is automatically $ C^{\omega}$, possibly in a smaller neighborhood of $ q$. This local result is an example which preceeds the theory developed by D. Kinderlehrer, L. Nirenberg and J. Spruck (1977-79), proving that in many cases free surfaces cannot have an arbitrary regularity; in particular, there exist $ k,\mu$ such that if the surface in question is $ C^{k,\mu}$, then automatically is $ C^{\omega}$. In this paper we extend their methods to Neumann type problems for free surfaces with surface tension.


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Additional Information

Walter Craig
Affiliation: Department of Mathematics and Statistics, McMaster University, Hamilton, Ontario, Canada L8S 4K1
Email: craig@math.mcmaster.ca

Ana-Maria Matei
Affiliation: Department of Mathematics and Computer Science, Loyola University New Orleans, 6363 St. Charles Avenue, New Orleans, Louisiana 70118
Email: amatei@loyno.edu

DOI: https://doi.org/10.1090/S0002-9939-07-08776-X
Received by editor(s): October 4, 2005
Received by editor(s) in revised form: March 20, 2006
Published electronically: March 14, 2007
Additional Notes: This research was supported in part by the Canada Research Chairs Program and the NSERC through grant # 238452-01.
Communicated by: Michael I. Weinstein
Article copyright: © Copyright 2007 American Mathematical Society
The copyright for this article reverts to public domain 28 years after publication.

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