Weighted a priori estimate in straightenable domains of local Lyapunov-Dini type

We prove the well-posedness of the Dirichlet problem for the Poisson equation in a weighted Sobolev space under weak assumptions both on the weight and on the boundary of the domain. The weight is supposed to satisfy the Muckenhoupt condition on the off-boundary cubes and an additional condition near the boundary. The boundary is Lipschitz, flat enough, straightenable (in a sense close to the one studied before by the author) and is either straightenable with small constant or satisfies the so-called local Lyapunov-Dini condition. The proof amounts to an a priori estimate obtained via localizing the problem, straightening the boundary, $L^p_w$-discretizing singular integrals and estimating a number of dyadic sums. Our results strengthen some of the results of V. G. Maz'ya, T. O. Shaposhnikova, K. Schumacher, R. G. Durán, M. Sanmartino and M. Toschi.