We consider elliptic equations of order 2m with Hölder coefficients. We show local solvability of the Dirichlet problem with m conditions on the boundary of the upper half space. First we consider local solvability in free space and then we treat the boundary case. Our method is based on applying the operator to an approximate solution and iterating in the Hölder spaces. A priori estimates for the approximate solution is the essential part of the paper.Facultad de Ciencias Exactas