A nonvariational form of the Neumann problem for Hölder continuous harmonic functions
Massimo Lanza de Cristoforis · Journal of Differential Equations · 2025
We present a nonvariational setting for the Neumann problem for harmonic functions that are Hölder continuous and that may have infinite Dirichlet integral. To do so, we introduce a space of distributions on the boundary. Namely, a space of first order traces for Hölder continuous harmonic functions that we characterize. Next, we analyze the properties of the distributional single layer potentials which are associated to a density in the space of first order traces and we prove a representation theorem for harmonic Hölder 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.