On the existence of foliations by solutions to the exterior Dirichlet problem for the minimal surface equation

Abstract

Given an exterior domain Ω of C 2,α class of R n , n ≥ 3, it is proved the existence of a foliation of the closure of an open set of R n+1 \ (∂Ω × R) which leaves are graphs of the solutions of the exterior Dirichlet problem on Ω for the minimal surface equation containing the trivial solution. The leaves have horizontal ends and are parametrized by the maximum angle α ∈ [0, π/2] that the Gauss map in R n+1 of the leaves make with the vertical direction at the boundary of the domain. The solutions have a limit c α at infinity, the map α → c α is increasing and bounded. Moreover, if R n \Ω satisfies the interior sphere condition of maximal radius ρ and if ∂Ω is contained in a ball of minimal radius ̺ thenwhereOne of the above inclusions is an equality if and only if ρ = ̺, Ω is the exterior of a ball of radius ρ and the solutions are radial. These foliations were studied by E. Kuwert in [6] and our result answers a natural question about the existence of such foliations which was not touched in [6].