We consider the mixed boundary value problem for linear second order elliptic equations in a plane domain lq whose boundary has corners, and obtain conditions sufficient for the solution to be in C2+(), where 0< a < 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.

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We study here the smoothness of solutions of the Dirichlet problem for elliptic equations in a region G with a piece-wise smooth boundary. The smoothness of the solution given depends on the smoothness of the coefficients of the equation, the boundary, the boundary function and the values of the angles on the boundary and the values of the coefficients of the second derivatives at the corners.

In this paper, smoothness properties of solutions of the initialDirichlet problem for parabolic equations in regions with edges are considered. We obtain bounds for solutions and derivatives, and prove the Holder continuity of the first derivatives and of the second derivatives multiplied by a suitable power of the distance from the edges.