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

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, 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, μ such that if the surface in question is C k,μ , then automatically is C ω . In this paper we extend their methods to Neumann type problems for free surfaces with surface tension.