Schauder estimate for solutions of Poisson's equation with Neumann boundary condition

TL;DR

This paper establishes existence and Schauder estimates for solutions to Poisson's equation with Neumann boundary conditions, filling a gap in the literature where such results were not explicitly proved.

Contribution

It explicitly proves the existence and provides Schauder estimates for the Neumann problem, which were previously not detailed in the literature.

Findings

Existence of solutions in Hölder spaces for Neumann problem

Schauder estimates for solutions with Neumann boundary conditions

Clarification of theoretical foundations for Neumann boundary problems

Abstract

In this work we consider the Neumann problem for the Laplace operator and we prove an existence result in the H\"older spaces and obtain Schauder estimates. According to our knowledge this result is not explicitly proved in the several works devoted to the Schauder theory, where similar theorems are proved in detail for the Dirichlet and oblique derivative problems. Our contribution is to make explicit the existence and the estimate for the Neumann problem.