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

Craig, Walter; Matei, Ana-Maria

doi:10.1090/S0002-9939-07-08776-X

On the regularity of the Neumann problem for free surfaces with surface tension
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Proc. Amer. Math. Soc. 135 (2007), 2497-2504 Request permission

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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