The regularity problem in domains with lower dimensional boundaries

In the present paper we establish the solvability of the Regularity boundary value problem in domains with lower dimensional boundaries (flat and Lipschitz) for operators whose coefficients exhibit small oscillations analogous to the Dahlberg-Kenig-Pipher condition. The proof follows the classical strategy of showing bounds on the square function and the non-tangential maximal function. The key novelty and difficulty of this setting is the presence of multiple non-tangential derivatives. To solve it, we consider a cylindrical system of derivatives and establish new estimates on the “angular derivatives”.