n this talk, we discuss two topics for the regularity theory of elliptic equations. First, we consider the Zaremba problem for local elliptic equations.
The problem allows rapidly alternating two boundary conditions such as the Dirichlet and the Neumann conditions.
We introduce higher integrability results and discuss further possible regularities. Next, we deal with Meyers' example of local elliptic equations and its application to nonlocal equations.