HADAMARD AND LIOUVILLE TYPE RESULTS FOR FULLY NONLINEAR PARTIAL DIFFERENTIAL INEQUALITIES * I. CAPUZZO DOLCETTA Dipartimento di Matematica, Universit` a di Roma - La Sapienza, 00185 Roma, Italy, e-mail: capuzzo@mat.uniroma1.it A. CUTR ` I Dipartimento di Matematica, Universit` a di Roma - Tor Vergata, 00133 Roma, Italy, e-mail: cutri@mat.uniroma2.it Abstract In this paper we prove some Hadamard and Liouville type properties for nonnegative viscosity supersolutions of fully non linear uniformly elliptic par- tial differential inequalities in the whole space. 1 Introduction In this paper we consider the fully nonlinear partial differential inequality F (x, u, Du, D 2 u) ≥ 0 , x ∈ IR N (1.1) with the aim of identifying sufficient conditions on the uniformly elliptic func- tion F which guarantee the validity of Liouville type results such as (A) any nonnegative solution of (1.1) is a constant or (B) the only nonnegative solution of (1.1) is u ≡ 0. This questions have been recently tackled in the framework of the theory of viscosity solutions (see [6] as a general reference on the subject), the natural one in view of the nonlinear dependence of F on second derivatives. The first property has been established for the equation F (D 2 u)=0asa consequence of the Krylov–Safonov–Harnack inequality for viscosity solutions in [3]. * This work was partially supported by the TMR Network ”Viscosity Solutions and Applications”. 1