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

TL;DR

This paper establishes an explicit equivalence between Dirichlet and Neumann problems for the Laplace operator in doubly-connected planar regions, extending previous results from simply connected regions and providing conditions for solution regularity.

Contribution

It demonstrates the equivalence of Dirichlet and Neumann problems in doubly-connected regions and offers conditions for the regularity of Neumann solutions.

Findings

Explicit formulas linking Dirichlet and Neumann solutions

Equivalence extends to doubly-connected regions

Conditions for higher order derivative regularity

Abstract

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.