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

Abstract

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