  In this paper we study the dependence of the H\"older estimates on the
geometry of a domain with holes for the Neumann problem. For this, we study the
H\"older regularity of the solutions to the Dirichlet and Neumann problems in
the disk (and in the exterior of the disk), from which we get a relation
between harmonic extensions and harmonic functions with prescribed Neumann
condition on the boundary of the disk (for both interior and exterior
problems).
