The Integral Equation Method and the Neumann Problem for the Poisson Equation on NTA Domains

The Neumann problem for the Poisson equation is studied on a general open subset G of the Euclidean space. The right-hand side is a distribution F supported on the closure of G. It is shown that a solution is the Newton potential corresponding to a distribution B ∈ε (clG), where ε(clG) is the set of all distributions with finite energy supported on the closure of G. The solution is looked for in this form and the original problem reduces to the integral equation TB = F. If the equation TB = F is solvable, then the solution is constructed by the Neumann series. The necessary and sufficient conditions for the solvability of the equation TB = F is given for NTA domains with compact boundary. The research was supported by the Academy of Sciences of the Czech Republic, Institutional Research Plan No. AV0Z10190503.