| We study the fully nonlinear elliptic equation F&parl0;D2u,Du, u,x&parr0;=f in a smooth bounded domain O, under the assumption the nonlinearity F is uniformly elliptic and positively homogeneous. Important examples of such operators include the Bellman operator, the Bellman-Isaacs operator, and the Pucci extremal operators. Recently, it has been shown that such operators have two principal "half-eigenvalues," and that the corresponding Dirichlet problem possesses solutions, if the principal half-eigenvalues are positive. We provide new proofs of these results, and generalize the ABP inequality to homogeneous operators with positive half-eigenvalues.;The Donsker-Varadhan minimax formula for the principal eigenvalue of a uniformly elliptic operator in nondivergence form is generalized to the first principal half-eigenvalue of a fully nonlinear operator which is concave (or convex), and positively homogeneous. 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.;The existence of solutions of the Dirichlet problem "between" the principal half-eigenvalues and the "second" eigenvalue is obtained, as well as an anti-maximum principle, generalizing the famous theorem of Clement and Peletier [22] for linear operators.;The last chapter, based on joint work with Maxim Trokhimtchouk, concerns the long-time behavior of solutions of the uniformly parabolic equation ut+F&parl0;D2u&parr0;=0 inRnx R+, where F is a positively homogeneous operator. We prove the existence of a unique positive solution phi + and negative solution phi-, which satisfy the self-similarity relations F+/-x,t =la+/-F+/- &parl0;l1/2x,lt&parr0;. We prove that the rescaled limit of any solution of the Cauchy problem with nonnegative (nonpositive) initial data converges to phi + (phi-) locally uniformly in Rnx R+ . The anomalous exponents alpha+ and alpha - are identified as the principal half-eigenvalues of a certain elliptic operator associated to F in Rn . |
