Mixed type boundary value problem of elliptic equation in a thin domain

In this paper, we prove the a priori estimates for two-dimensional second order homogeneous linear elliptic equations in a narrow region. In a crescent-shaped area, part of the boundary is subject to an oblique derivative boundary condition, while the other part of the boundary is subject to a Dirichlet boundary condition. We show that, as the crescent-shaped area collapses into a segment under suitable conditions, the boundary value problem obeys uniform Schauder estimates and induces an asymptotic estimate.