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ISSN 1088-6826(e) ISSN 0002-9939(p)

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

Author(s): Walter Craig; Ana-Maria Matei
Journal: Proc. Amer. Math. Soc. 135 (2007), 2497-2504.
MSC (2000): Primary 35J65, 76B15
Posted: 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 in a neighborhood of a point situated on the free surface is automatically $C^{\omega}$ , possibly in a smaller neighborhood of . 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.

References:

1.: S. Agmon, A. Douglis, & L. Nirenberg. Estimates near the boundary for solutions of elliptic partial differential equations satisfying general boundary conditions: I. Comm. Pure Appl. Math. 12 (1959), 623-727. MR 0125307 (23:A2610)
2.: C. J. Amick, L. E. Fraenkel, & J. F. Toland. On the Stokes conjecture for the wave of extreme form. Acta Matematica 148 (1982), 193 - 214. MR 0666110 (83m:35147)
3.: W. Craig & A.-M. Matei. Sur la régularité des ondes progressives à la surface de l'eau. Journées Equations aux Dérivées Partielles, Exp. No. IV, 9 pp., Univ. Nantes, Nantes, 2003. MR 2050590 (2004m:35209)
4.: D. Gilbarg & N. S. Trudinger. Elliptic Partial Differential Equations of Second Order, Springer Verlag, New York - Berlin, (1977). MR 0473443 (57:13109)
5.: D. Kinderlehrer, L. Nirenberg. Regularity in free boundary value problems. Ann. Scuola. Norm. Sup. Pisa, Ser IV, IV (1977), 373-391. MR 0440187 (55:13066)
6.: D. Kinderlehrer, D. L. Nirenberg & J. Spruck. Regularity in elliptic free boundary problems I. Journal d'Analyse Math. 34 (1978), 86-119. MR 0531272 (83d:35060)
7.: D. Kinderlehrer, L. Nirenberg & J. Spruck. Regularity in elliptic free boundary problems II: Equations of higher order. Ann. Scuola. Norm. Sup. Pisa, Ser IV VI (1979), 637-683. MR 0563338 (83d:35061)
8.: H. Lewy. A note on harmonic functions and a hydrodynamical application. Proc. AMS 3 (1952), 111-113. MR 0049399 (14:168c)
9.: A.-M. Matei. The Neumann problem for free boundaries in two dimensions. C.R. Acad. Sci. Paris, Ser. I 335 (7), (2002), 1-6. MR 1941301 (2003i:35286)
10.: C. B. Morrey. Multiple integrals in the calculus of variations. Springer Verlag, New York - Berlin, (1966). MR 0202511 (34:2380)

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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: 10.1090/S0002-9939-07-08776-X
PII: S 0002-9939(07)08776-X
Received by editor(s): October 4, 2005
Received by editor(s) in revised form: March 20, 2006
Posted: 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
Copyright of article: Copyright 2007, American Mathematical Society
The copyright for this article reverts to public domain after 28 years from publication.

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