Boundary estimates for solutions of the Monge–Ampère equation satisfying Dirichlet–Neumann type conditions in annular domains

Abstract

We consider smooth solutions of the Monge–Ampère equation on an annular domain, whose boundary consists of two smooth, closed, strictly convex hypersurfaces, subject to mixed boundary conditions. In particular we impose a homogeneous Dirichlet condition on the outer boundary and a Neumann condition on the inner boundary. We demonstrate that in general, global C2 estimates cannot be obtained unless we impose extra restrictions on the principal curvatures of the inner boundary and on the Neumann condition itself. The latter is illustrated by the construction of an explicit counterexample. Under these conditions, we prove a priori C2 estimates and show that our problem admits a smooth solution.