On Solutions of Elliptic Equations Satisfying Mixed Boundary Conditions

Abstract

We consider the mixed boundary value problem for linear second order elliptic equations in a plane domain $\Omega $ whose boundary has corners, and obtain conditions sufficient for the solution to be in $C^{2 + \alpha } (\bar \Omega )$, where $0 < \alpha < 1$. This result means that under those conditions, solutions are as smooth as they would be in the absence of corners, so that, in this sense, the present result is best possible.