  Motivated by recent results regarding the equivalence of the Dirichlet and
Neumann problems for the Laplace operator in the case of simply connected
regions, the present paper takes a step further and provides a similar
equivalence between the above mentioned problems in the case of planar
doubly-connected regions. The equivalence means that solving any of these
problems leads by an explicit formula to a solution of the other problem. In
addition, suffcient conditions guaranteeing the uniform Holder continuity of
the higher order partial derivatives of the solutions to the Neumann problem
are provided as a consequence of this equivalence.
