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

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,C-mu, then automatically is C-omega In this paper we extend their methods to Neumann type problems for free surfaces with surface tension.