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

Equivalence of the Dirichlet and Neumann problems for the Laplace operator in planar doubly-connected regions