In this paper, the maximum principle for some n-dimensional coercive Dirichlet problem of the second order is proved and sufficient conditions for the existence of an optimal solution are given. The results obtained generalize, in the sense of the dimension of the state space, some special case of the maximum principle for the one-dimensional Dirichlet problem, derived in [M. Goebel and V. Raitums, J. Global Optim., 4 (1994), 367-395].