We study the regularity of the solution of Dirichlet problem of Poisson equations over a bounded domain.A new sufficient condition,uniformly positive reach is introduced.Under the assumption that the closure of the underlying domain of interest has a uniformly positive reach,the H2 regularity of the solution of the Poisson equation is established.In particular,this includes all star-shaped domains whose closures are of positive reach,regardless if they are Lipschitz domains or non-Lipschitz domains.Application to the strong solution to the second order elliptic PDE in non-divergence form and the regularity of Helmholtz equations will be presented to demonstrate the usefulness of the new regularity condition.