The regularity problem with a weaker condition on only the transversal direction

We study an elliptic operator L: =div (A) on the upper half space. It is known that if the matrix A is independent in the transversal t-direction, then the regularity boundary value problem is solvable with data in a Sobolev space. In the present paper we improve on the t-independence condition by introducing a mixed L¹-L^ condition that only depends on ₜ A, the derivative of A in transversal direction. This condition is different from other conditions in the literature and has already been proven to imply solvability of the Dirichlet boundary value problem.