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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

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.