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.
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Massimo Lanza de Cristoforis (2024) studied this question.
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