Invertibility of layer potential operators on Besov spaces and Poisson's equation for the Laplacian in Lipschitz domains

We study global regularity for Poisson's equation for the Laplacian with homogeneous Neumann boundary condition, when the right-hand side belongs to a Sobolev space {dollar}Lsbsp{lcub}s-20{rcub}{lcub}p{rcub}(Omega){dollar} (the closure of {dollar}Csbsp{lcub}0{rcub}{lcub}infty{rcub}(Omega){dollar} in {dollar}Lsbsp{lcub}s-2{rcub}{lcub}p{rcub}(Omega){dollar} and {dollar}Omega{dollar} is a bounded Lipschitz domain in {dollar}IRsp{lcub}n{rcub}{dollar}. We obtain an optimal range for s and p for which the problem is solvable in the class {dollar}Lsbsp{lcub}s{rcub}{lcub}p{rcub}(Omega){dollar} and natural estimates in terms of the Sobolev norms hold. We settle this question by studying the mapping properties of the single and double layer potentials on the scale of Besov spaces on the boundary of {dollar}Omega{dollar} and the invertibility of the corresponding boundary operators on the same scale. Incidentally, we recover a recent result by Jerison and Kenig, regarding the same regularity question for the Dirichlet problem. Our results can be applied to obtain the L{dollar}sp{lcub}p{rcub}{dollar} continuity of the Helmholtz decomposition of vector fields defined on {dollar}Omega{dollar}. Our methods do not depend on the maximum principle, and hence there is no obstacle to extend our results to certain three dimensional elliptic systems.