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

H\"older estimates for the Neumann problem in a domain with holes and a relation formula between the Dirichlet and Neumann problems