A nonvariational form of the Neumann problem for the Poisson equation

TL;DR

This paper introduces a nonvariational approach to the Neumann problem for the Poisson equation, extending the concept of normal derivatives and solving for solutions with less regularity and infinite Dirichlet integrals.

Contribution

It develops a new nonvariational framework for the Neumann problem, including a distributional boundary normal derivative for less regular solutions.

Findings

Extended normal derivative concept for harmonic functions

Solved Neumann problem with distributional boundary data

Applicable to solutions with infinite Dirichlet integral

Abstract

We present a nonvariational setting for the Neumann problem for the Poisson equation for solutions that are H\"{o}lder continuous and that may have infinite Dirichlet integral. We introduce a distributional normal derivative on the boundary for the solutions that extends that for harmonic functions that has been introduced in a previous paper and we solve the nonvariational Neumann problem for data in the interior with a negative Schauder exponent and for data on the boundary that belong to a certain space of distributions on the boundary.