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 $C^{k,\mu}$, then automatically is $C^{\omega}$. In this paper we extend their methods to Neumann type problems for free surfaces with surface tension.