HOLDER ESTIMATES FOR THE NEUMANN PROBLEM IN A DOMAIN WITH HOLES AND A RELATION FORMULA BETWEEN THE DIRICHLET AND NEUMANN PROBLEMS
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2020
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Abstract
In this paper we study the dependence of the Holder estimates on the geometry of a domain with holes for the Neumann problem. For this, we study the Holder 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). A novelty of this work is that we deal directly with the Holder regularity of the single layer potentials of the Dirichlet and Neumann problems for the Poisson equation, something that most of the times seems to be avoided by studying the Newtonian potential.
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Dirichlet problem, Neumann problem, elliptic regularity