  We generalize the Donsker-Varadhan minimax formula for the principal
eigenvalue of a uniformly elliptic operator in nondivergence form to the first
principal half-eigenvalue of a fully nonlinear operator which is concave (or
convex) and positively homogeneous. Examples of such operators include the
Hamilon-Jacobi-Bellman operator and the Pucci extremal operators. In the case
that the two principal half-eigenvalues are not equal, we show that the
measures which achieve the minimum in this formula provide a partial
characterization of the solvability of the corresponding Dirichlet problem at
resonance.
