A nonvariational form of the Neumann problem for H\"{o}lder continuous harmonic functions

Abstract

We present a nonvariational setting for the Neumann problem for harmonic functions that are H\"{o}lder continuous and that may have infinite Dirichlet integral. Then we introduce a space of distributions on the boundary (a space of first order traces for H\"{o}lder continuous harmonic functions), we analyze the properties of the corresponding distributional single layer potential and we prove a representation theorem for harmonic H\"{o}lder continuous functions in terms of distributional single layer potentials. As an application, we solve the interior and exterior Neumann problem with distributional data in the space of first order traces that has been introduced.